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THE GLASGOW TEXT BOOKS OF CIVIL 

ENGINEERING. Edited by G. MONCUR, B.Sc. 

M.I.C.E. Professor of Civil Engineering in the Royal 

Technical College, Glasgow. 



SURVEYING & FIELD WORK 



THE GLASGOW TEXT BOOKS. 
Edited by G. MONCUR. 



SURVEYING & FIELD WORK 



A PRACTICAL TEXT-BOOK ON SURVEYING 
LEVELLING & SETTING-OUT 



Intended for the Use of Students in Technical Schools 

and Colleges and as a Work of Reference for Surveyors, 

Engineers and Architects 



BY 

JAMES WILLIAMSON 
A.M.Inst.C.E. 




NEW YORK 

D. VAN NOSTRAND COMPANY 

25 PARK PLAGE 

1915 



4 



// 



LC Control Number 



tmp96 025489 



Printed tn Great Britian 



PREFACE 



The aim of the author in the preparation of this work has been 
to produce a thoroughly sound text-book on the principles and 
practice of surveying, including levelling and setting-out, which, 
while being primarily intended for the use of students in technical 
schools and colleges, would at the same time be of value as a work 
of reference for the surveyor, engineer and architect. 

In the arrangement of the work the time-honoured plan of 
beginning with chain surveying, and making it the vehicle for a 
comprehensive description of the elements of field work and office 
work, has been adhered to. While it may appear to some that too 
great importance is thereby attached to chain surveying, the 
arrangement has the advantage of taking the beginner at once, 
in the first eight chapters of the book, through all the processes 
which go to the attainment (by particular methods) of the ultimate 
purpose of surveying, namely, the finished survey plan. The 
various methods of surveying involving measurement of angles are 
dealt with in subsequent chapters, the order of consideration in 
each case being first the field work, then the office work. Traverse 
surveying with the theodolite has been somewhat fully dealt with, 
as being probably the method in most general use for surveying 
limited areas, and very full consideration has been given to the co- 
ordinate or latitude and departure method of plotting traverse 
surveys. The subject of minor triangulation has been included 
because of its importance in the case of surveys which extend over 
larger areas, but the subjects of tacheometry, plane tabling, 
geodetic surveying and astronomical work, and the more difficult 
branches of levelling and setting-out, have not been dealt with 
in this volume. 

Endeavour has been made, by means of worked-out examples, 
and the employment of over 270 diagrams, all specially prepared 
for this work, to make the explanations of principles and methods 
and descriptions of instruments clear and explicit. In regard to 



vi PREFACE 

field work and surveying practice, and the office work of preparing 
plans, numerous practical hints have been given, drawn from a wide 
experience, and attention has been called to many of the pitfalls 
which beset the beginner. 

The necessity for accuracy and precision in all surveying opera- 
tions and the importance of systematic routine and proper checking 
have been emphasised throughout. Absolute accuracy in the 
measurement of lengths and angles being, however, out of the 
question, the aim in practical surveying, where the time and money 
available for the work are usually both strictly limited, should be to 
attain in the most economical manner at least that degree of pre- 
cision which is necessary for the purpose of the survey. Precision 
in surveying is attained by a process of cutting down errors, and, 
in the author's opinion, it is only from a careful appreciation of the 
relative magnitudes of the errors which may arise from various 
sources in the operations of surveying and levelling that a sur- 
veyor is enabled to expend his care and effort to the best advantage. 
For this reason the subject of errors has throughout received 
detailed treatment. 

The author's acknowledgments are due to Mr. Guthrie Brown 
for valuable assistance in the preparation of the diagrams, to 
Messrs. Cooke for the loan of the block from which Fig. 127 was 
prepared, and to many authors in the field of surveying for 
suggestions. 

JAMES WILLIAMSON. 



TABLE OF CONTENTS 



CHAPTER I 

Surveying — Fundamental Principles 

page 

Use of Survey Plans 1 

Methods of representing Surface of Ground on Reduced Scale . 1 

Plane Surveying 1 

" Surveying," what it includes ....... 2 

Surveying based on simple Geometrical Principles .... 2 

Four Methods of fixing Relation of a Point to two known Points 2 

Methods applied to determining Altitudes of Points . . . 5 

Effect of Curvature of Earth 6 

As to neglecting Curvature of Earth ...... 7 

Curvature of Earth to be considered in finding Altitudes . . 8 

Horizontal Projections of Lengths and Angles to be Measured . 9 

Difficulties of Surveying not in Principles but in their Application. 9 

CHAPTER II 

Chain Surveying — Instruments 

British Units of Linear Measurement . . . . 10 

Metrical Units of Measurement . . . . . . .11 

Measuring Instruments : — Chain, Steel Band, Etched Steel Band, 
Steel Tape, Linen Tape, Offset Staff, Wooden Measuring 
Rods, Metre Chain, Arrows, Ranging Poles, Flags, Laths 
and Whites, Pegs . . . . . . . . .11 

Cross Staff 15 

Optical Square ... 16 

Use of Optical Square 18 

Use of Optical Square on Sloping Ground . . . . .19 

Testing the Optical Square . . . . . . . . 19 

Prism Square .......... 20 

Wooden Set-square . . . . . . . . . . 21 

Line Ranger 21 

Inclinometer ........... 22 

Abney's Level 23 

The Human Eye 24 

CHAPTER III 

Chain Surveying — Field Operations 

Survey Party and Equipment . . . . . . .26 

Ranging Lines ........... 26 

Ranging Line across a Hollow . . . . . . .29 



viii TABLE OF CONTENTS 

PAGE 

Ranging Line over a Hill ......... 30 

Ranging past Obstacles . . . . . . . .31 

Chaining Survey Lines . . . . . . . . . 31 

Field Standard . . 32 

Chaining on Level Ground ........ 33 

Precautions to be observed by Leader and Follower ... 35 

Chaining on Sloping Ground — Three Methods 36 

CHAPTER IV 

Chain Surveying — Running a Survey Line 

Fixing Positions of Objects relative to Points on a Survey Line . 40 

Points to which Offsets or Ties should be taken .... 43 

Fixing Buildings, &c. ......... 44 

Locating Irregular Boundaries, &c 47 

Length of Offsets 48 

Running a Survey Line ........ 48 

Field Book and Field Notes 49 

Examples of Survey Lines .52 

CHAPTER V 

Chain Surveying — Arrangement of Survey Lines 

Arrangement of Survey Lines . . . . . . . . 55 

Marking Survey Stations ........ 59 

Referencing Survey Stations 59 

Numbering Stations 60 

CHAPTER VI 

Chain Surveying — Errors 

Errors in measuring a Length . . . . . . . . 61 

Errors in Length of Chain ........ 61 

Errors from Chain or Tape not being Stretched Horizontal . . 63 
Errors from Chain not being Stretched Straight and from Sag of 

Chain 63 

Errors in Fixing Arrows and Marking Chain Lengths . . . 64 

Errors and Mistakes in Reading Chain or Tape .... 64 

Errors in Ranging Out Survey Lines . . . . . . 65 

Errors in Locating Objects from Survey Lines .... 65 

Permissible Error .......... 66 

Expedition in Surveying ........ 67 

CHAPTER VII 

Chain Surveying — Special Problems 

Setting Out a Right Angle 69 

Dropping a Perpendicular from a Point on to a Chain Line . . 71 

Setting Out a Parallel Line . . . . . . . . 71 

Setting Out a Given Angle : — Tangent Method, Sine Method, 

Method by Sine of Half the Angle 72 

Overcoming Obstacles to Chaining . . . . . . . 72 



TABLE OF CONTENTS ix 

PAGE 

Chaining Past an Inaccessible Area . . . . . .73 

Chaining Past an Obstruction which cannot be seen through — 

Three Methods 75 

Distance Across a River 77 

Surveying Far Side of Eiver . . . . . . 78 

Surveying Pond, Wood, &c. . . 78 

Chain Surveying in Towns . . . . . 80 

CHAPTER VIII 

Plotting the Plan 

Drawing Instruments, &c. . . . . . . . .81 

Commencing the Plan ......... 88 

Arranging of Plan on the Sheet 88 

Laying Off the Base Line 89 

Plotting the Main Triangles 89 

Plotting Offsets 90 

Plotting Ties 91 

Plotting Details 91 

Errors in Plotting .91 

Pencilling, Inking-in, Erasing ........ 92 

Conventional Signs : — Full Black Lines, Dotted Lines, Coloured 

Lines ........... 93 

Fences and Boundaries . . . 94 

Boads ; Railways .......... 94 

Buildings 97 

Various Kinds of Land ........ 97 

Example from Survey Plan 97 

Printing and Lettering . . . . . . . .97 

Colouring and Tinting 99 

CHAPTER IX 

'- Compass and Sextant Surveying 

The Magnetic Needle . . . 101 

Magnetic North and Magnetic Meridian 102 

Magnetic Declination 103 

Local Attraction 103 

Bearing of a Line 104 

Whole Circle Bearing or Azimuth 105 

Forward Bearing and. Back Bearing . . •, ■ • • • ^^ 
Surveyor's Compass . . . . . . . . . 106 

Prismatic Compass 108 

Compass on Theodolite . . . . . . . .109 

Methods of Graduating the Compass Card 109 

Taking a Magnetic Bearing with the Compass . . . .110 

Advantages and Limitations of the Compass . . . ..Ill 

Use of Compass Surveying . . . . . . . .111 

Compass Surveying with Needle only (Free Needle) . . . . 112 

Correction for Local Attraction 113 

Booking the Survey . . . . . . . . . . 114 

Surveyor's or Box Sextant . . 114 

Testing the Sextant 117 

Advantages and Limitations of the Sextant . . . .118 



x TABLE OF CONTENTS 

CHAPTER X 

The Theodolite 

page 

Theodolite 120 

Support or Stand 122 

Parallel Plates 122 

Three-screw Levelling Arrangement . . . . . .123 

Graduated Horizontal Circle . . . . . . . . 124 

Vernier Plate 124 

Standards, Telescope, Vertical Circle, &c. . . . . . 125 

Telescope 126 

Vernier 128 

Theodolite Verniers . . . . . . . . .130 

Use of Theodolite ; Setting up the Theodolite 133 

Setting over a Point . . . . . . . . .134 

Levelling up 137 

Measuring Horizontal Angles . . . . . . .137 

Measuring a Vertical Angle . . . . . . . 139 

Measuring Angles by Repetition . . . . . .140 

Ranging a Straight Line . . . . . . . . . 140 

Prolonging a Line 141 

Ranging a Straight Line between Stations which are Invisible from 

each other 142 



CHAPTER XI 

Traverse Surveying with the Theodolite 

Traverse Surveying with the Theodolite . . . . .143 

Laying out Traverse Survey Lines . . . . . . . 144 

Fixing Survey Stations . . . . . . . .145 

Methods of Reading Angles . 145 

Direct Angle or Separate Angle Method . . . . .145 

"Whole Circle Bearing Methods 148 

Comparison of Methods of Measuring Whole Circle Bearings . . 151 
Booking Angles of Traverse Survey 151 

CHAPTER XII 

Plotting a Traverse Survey by Angle and Distance 

Methods of Plotting 153 

Laying off a Single Angle or Bearing ; Protractor, Tangent, and 

other Methods 154 

Direct or Separate Angle Method of Plotting Traverse Survey 
Lines : — (a) Using the Protractor ; (b) Using the Tangent 

Method 156 

Whole Circle Bearing Method of Plotting Traverse Survey 
Lines : — (a) Using the Protractor ; {b) Using the Tangent 

Method 158 

Comparison of Angle and Distance Methods .... 162 

Checks on Unclosed Traverse 162 

Graphical Adjustment of Closing Error . . . . .164 



TABLE OF CONTENTS 



CHAPTER XIII 

Plotting Traverse Survey by Co-ordinate or Latitude and 
Departure Method 

page 
Co-ordinate Method of Plotting Traverse Survey . . . .166 

Latitude and Departure ; Co-ordinates 167 

Distinctions between North. Latitude and South Latitude ; East 

Departure and West Departure 168 

Calculation of Latitudes and Departures 169 

Calculation of Co-ordinates . . . . . . 172 

Adjustment of Angular Errors 175 

Adjustment of Latitudes and Departures ..... 176 

Weighting the Survey 177 

Graphical Method of Correcting for Errors in Chainage only . 177 

Plotting Survey Stations by Co-ordinates . . .... 180 

Alteration of Bearings to suit new Co-ordinate Axes . . . 182 

Connecting Survey Lines from one Sheet to another . . . 183 

Closing Error and Limits of Error .184 

CHAPTER XIV 

Triangulation 

Triangulation ; Purpose of Triangulation 186 

Precautions to Ensure Accuracy . 188 

Calculation of Lengths of Sides of Triangles 188 

Closing Error in Triangulation 190 

Field Work 190 

Measuring the Base Line 190 

Broken Base Line ; Enlarging a Base Line . . . . .192 

Selecting Triangulation Stations ; Marking Stations . . . 195 
Measurement of Angles ; Repetition Method ; Series or Reiteration 

Method . . 198 

Adjustment of the Angles . . . . • . . . 201 

Plotting Triangulation Stations 202 

Errors in Reading Angles 203 

Error in Planting the Theodolite 204 

Error from Incorrect Levelling ; Incorrect Focussing . . . 204 

Error from Incorrect Bisection ; Displacement of Signal . . 205 

Error from Natural Causes 205 

Error from Incorrect Adjustment of Line of Collimation . . 206 

Error from Incorrect Adjustment of Horizontal Axis . . . 207 

Error from Incorrect Graduation, &c. . . . . . . 207 

CHAPTER XV 

Some Survey, Traverse, and Triangulation Problems 

Perpendiculars 208 

Setting out a Line Parallel to a given Line ..... 209 

Running a Line between two Points when an Obstacle Intervenes 209 

Distance to an Inaccessible Point . 210 

Distance between two Inaccessible Points . . . . .211 
Distance of a Boat from the Shore ..211 



xii TABLE OF CONTENTS 

PAGE 

Given the Latitude and Departure of a Line, to find its Length and 

Bearing 212 

Given the Co-ordinates of two Points, to find the Length and Bear- 
ing of the Line joining them 213 

To find the Length and Bearing of an Omitted Side of a Traverse 

Polygon 213 

To find where a Survey Line cuts a Parallel to one of the Axes . 214 
Two Omitted Measurements in a Polygon . . . . .216 

Three-point Problem 217 

Determination of Heights by the Theodolite . . . .219 

Trigonometric Levelling 220 

To find Difference of Altitude between two Points . . .221 

CHAPTER XVI 

Levelling 

Levelling 224 

Level Surface 224 

Water Level ; Spirit Level 225 

Mechanic's Level 227 

Dumpy Level 228 

Troughton & Simms Level ........ 233 

Wye Level 233 

Staff .233 

Use of Level and Staff 236 

Signals 238 

Datum ; Bench Mark ; Ordnance Datum 239 

Calculation of Levels . . . . . . . . 240 

Continuous Levelling 242 

Booking the Readings, and Reducing the Levels .... 245 

CHAPTER XVII 

Errors in Levelling 

Faulty Adjustment of Level 251 

Mistakes and Carelessness in Use of Level .... 252 

Errors resulting from Staff and its Manipulation . . . 252 
Inaccuracies and Mistakes in Reading the Staff and Booking the 

Readings ........... 254 

Curvature, Refraction and other Natural Sources . . . 255 

Mistakes in Reducing the Levels . . . . . . . 257 

Appropriate Length of Sight 257 

Permissible Error in Levelling ........ 258 

CHAPTER XVIII 

Sections, Contours, &c. 

Taking Longitudinal and Cross Sections 260 

Example of Section for Small Sewer 262 

Example of Longitudinal Section for Railway .... 263 

Cross Sections 267 

Contours ; Use of Contours . . . . . . .271 



TABLE OF CONTENTS xiii 

PAGE 

Locating Contours 276 

Use of Contour Plans 280 

Levelling on Steep Slope 284 

Levelling over Summits and Hollows 284 

Taking Level of Overhead Point 285 

Obstructions to Levelling : — Close Boarded Fence, Wall, Pond, &e. 286 

Keciprocal Levelling ......... 288 

Contour Grading 289 

CHAPTER XIX 

Setting Out Curves, &c. 

Methods requiring Chain, Tape and Optical Square . . . 292 

Method by Offsets Scaled from a Plan 292 

Method by Radius Swung from Centre of Curve .... 293 

Method by Offsets from a Tangent Line 293 

Method by Offsets from Chords produced ..... 297 

Curve Problems 299 

Setting out Curves with Theodolite 301 

Curve to Left 307 

Inaccessible Intersection Point 307 

Obstructions in Setting out Curves ....... 309 

Setting out Building Work 309 

CHAPTER XX 
Calculation of Areas 

Square Measure 313 

Areas of Geometrical Figures 314 

Area of Land .......... 316 

Methods of Taking out Areas 316 

Areas from Survey Plan . . . . . . . .316 

By Dividing Area up into Geometrical Figures . . . . 316 

By Dividing Area into Parallel Strips 318 

By Dividing Area into Squares ....... 320 

Areas by means of Offsets 320 

Areas by Planimeter . 322 

Areas by Direct Calculation from Field Measurements . . .325 

Area of a Traverse . 326 

Correction for Shrinkage of Plan 327 



CHAPTER XXI 

Calculation of Earthwork Quantities 

Purpose of Earthwork Calculations 330 

Volumes of Solid Bodies 331 

Prismoidal Formula 333 

Excavation in Foundation Pit 334 

Excavation and Embankment for Pond 336 

Quantity of Earthwork in Levelling an Area 337 

Earthwork for Roads, Railways, &c ■ 339 



xiv TABLE OF CONTENTS 

CHAPTER XXII 

Adjustment of Instruments 

page 

Dumpy Level ........... 343 

Peg Method of Adjustment to make the Line of Sight Parallel 

to the Bubble Axis 343 

To make the Line of Sight and Bubble Axis Perpendicular to 

the Vertical Axis 345 

Adjustment when Bubble is permanently fixed to Telescope . 346 
Adjustment when the Telescope is firmly fixed to the Vertical 

Axis 346 

Wye Level 347 

Adjustment of the Cross Hairs 347 

To make the Axis of the Bubble Tube Parallel to the Line of 

Supports of the Wyes 348 

To make the Axis of the Bubble Tube Perpendicular to the Vertical 

Axis of the Instrument ........ 348 

Theodolite 348 

Adjustment of the Plate Levels 349 

Adjustment of Cross Hairs . . . . . . . .349 

Adjustment of the Supports at Top of the Standards . . . 351 
Adjustment of the Axis of the Telescope Level Parallel to the 

Line of Sight 351 

Adjustment of Vertical Circle Verniers . ..... 351 

APPENDIX 

Geometric and Trigonometric Formula 

Right Angle Triangle 353 

Oblique Triangles 353 

Solution of Oblique Triangles ........ 354 



Index 355 



LIST OF 1LLUSTEATIONS 

FIG. ■ PAGE 

1. Fixing a Point by One Linear Measurement ... 3 

2. Fixing a Point by Two Linear Measurements . . . 3 

3. Fixing a Point by One Linear and One Angular Measure- 

ment .......... 4 

4. Fixing a Point by Two Angular Measurements . . . 4 

5. Measurement of Altitude. Method No. 3 . . . . 6 

6. Curvature of Earth . . . . ... . • ■ 7 

7. Measuring Horizontal Projections of Lengths and Angles , 8 

8. Divisions of Chain ......... 13 

Detail of Chain 13 

Divisions of Steel Band . . . . . . . 13 

Steel Band .13 

Cloth Tape . . 13 

Cloth Tape, Alternative Marking . . . . .13 

Cross Staff . . 15 

Principle of Optical Square . . . . . .16 

Optical Square . . . . . . . . . 17 

Section of Optical Square . . . - . . .17 

Use of Optical Square . . . . . . . . 18 

Stand for Use of Optical Square 18 

Setting out a Eight Angle on Sloping Ground . . . 19 

Testing Optical Square ....... 20 

Principle of Prism Square 20 

Principle of Five-sided Prism . . . . .20 

Zeiss Prism Square 21 

Principle of Line Banger 22 

Inclinometer . . . . . . . . . 22 

Use of Inclinometer ........ 23 

Principle of Abney's Level . . . . . . . 23 

Field of View of Abney's Level 23 

Banging Line across a Hollow . . . . . . 29 

Device in Banging Line 30 

Banging Line over a Hill 30 

30 

S 6 



xvi LIST OF ILLUSTRATIONS 

FIG. PAGE 

34. Official Standard for Testing Chain 31 

35. Field Standard for Testing Chain 32 

36. Measuring on Sloping Ground . . . . . . . 36 

37. Offsets 41 

38. Ties . 41 

39. Finding Correct Offset Distance ..... 42 

40. Accurate Short Offsets 42 

41. Offsets to Boundary ........ 43 

42. Fixing a Building 44 

43. Complete Measurements to fix a Building . ... 44 

44. Fixing Building from 'Short Side 45 

45. Fixing Building from Long Side 45 

46. Fixing Irregular Buildings . . . . . . . 46 

47. „ 46 

48. Offsets to Irregular Boundary. . . . . . . 47 

49. Field Book 49 

50. Field Book, to open Long . . . . . . . 49 

51. Example of Survey Line . . . . . . .51 

52. Example of Survey Line — Railway Lines . . . . 53 

53. Example of Survey Line — stream, etc. .... 54 

54. Displacement of a Point due to Error in Side . . . 55 

55. Limits of Well-conditioned Triangle ..... 55 

56. Proof Line for Triangle 56 

57. Single Triangle forming Basis of Survey Lines ... 57 

58. Survey Lines for Quadrilateral Enclosure . . . . 58 

59. „ „ „ 58 

60. Arrangement of Triangles 59 

61. Error from Careless Fixing of Arrow ..... 64 

62. Setting out Right Angle 69 

63. „ 69 

64. „ „ 69 

65. Right Angle by Steel Band 70 

66. Perpendicular from a Point 70 

67. „ „ 70 

68. Setting out a Parallel Line . . . . . . . 71 

69. „ „ 71 

70. Setting out an Angle. Tangent Method . . . . 71 

71. Setting out an Angle. Sine Method 72 

72. Setting out an Angle. Chord Method 72 

73. Chaining past Pond, etc 73 



LIST OF ILLUSTRATIONS xvii 

FIG. PAGE 

74. Chaining past and Surveying Pond 73 

75. Chaining past Pond, etc. ....... 74 

76. Chaining past Obstruction . . . . . . . 74 

77. Prolonging Chain Line past Obstruction .... 75 

78. „ „ „ „ .... 75 

79. „ „ „ „ .... 76 

80. Distance across a Kiver . . . . . . . . 76 

81. „ „ 76 

82. „ „ 77 

83. Surveying Far Side of River 78 

84. Surveying Wood, etc. . . . . . . . . 79 

85. Surveying Pond, etc. . . . . . . . .79 

86. Chain Surveying in Towns 80 

87. Set-squares 83 

88. Testing Set-square 84 

89. Testing Straight-Edge . . . • . . ' . .84 

90. Pen Spring Bows 86 

91. Pencil Spring Bows 86 

92. Pen and Pencil Bow Compasses 86 

93. Dividers 86 

94. Beam Compasses 86 

95. Manufactured Curves . . . . . . . .86 

96. Parallel Ruler 87 

97. Drawing Pen 88 

98. Laying off Long Distance . . . . . . . 89 

99. Plotting Offsets, etc 91 

Conventions for Fences and Boundaries . . . . 95 

Conventions for Roads ....... 95 

Conventions for Railways . . . . . . . 95 

Conventions for Buildings . . . .* . .96 

Fir Wood . . . 96 

Mixed Wood . 96 

Orchard 96 

Moorland 96 

Marsh and Pond 96 

Cliffs and Shore 96 

Plate I. — Example from Survey Plan . . Facing 96 

Plate II. — Examples of Lettering . . . „ 98 

110. Broad Magnetic Needle 102 



xviii LIST OF ILLUSTRATIONS 

FIG. PAGE 

111. Edge Bar Needle 102 

112. Diagram of Secular Variation of Magnetic Declination at 

London ........-• 104 

113. Bearing of a Line 105 

114. Bearings of Lines 105 

115. Forward and Back Bearing 106 

116. Forward and Back Bearing (Whole Circle) . . . . 106 

117. Forward and Back Bearings of Lines ..... 107 

118. Surveyor's Compass ........ 108 

119. Graduation of Prismatic Compass 109 

120. Explaining Graduation of Prismatic Compass . . . 109 

121. Graduations of Compass on Theodolite . . . .110 

122. Compass Bearings of a Polygon 113 

123. Principle of Box Sextant ....... 115 

124. Top View of Box Sextant 116 

125. Horizontal Section of Box Sextant 117 

126. Measuring Large Angle with Sextant . . . . . 118 

127. Theodolite 121 

128. Triangular Stand 123 

129. Telescope of Theodolite 126 

130. Diaphragm and Cross Hairs 128 

131. Vernier Reading to Tenths 129 

132. Vernier Reading to Minutes . . . . . . . 131 

133. Vernier Reading to 20 Seconds . . . . . .133 

134. Suspension of Plumb-bob . . . . . . . . 135 

135. Measuring a Horizontal Angle ...... 138 

136. Prolonging a Line 141 

137. Unclosed Traverse 143 

138. Single Closed Traverse 144 

139. Traverse Network 144 

140. Reading Exterior or Interior Angles 146 

141. Calculation of Bearings from Separate Angles . . .146 

142. Whole Circle Bearings by Direct Bearing Method . . . 149 

143. Plotting Angle by Tangent Method 155 

144. Plotting Angles over 45° 155 

145. Plotting Bearings by Tangent Method .... 156 

146. Plotting Angle by Chord Method 156 

147. Plotting Survey Lines by Protractor ..... 157 

148. Plotting Survey Lines by Tangent Method . . . . 157 

149. Plotting Whole Circle Bearings by Protractor . . . 159 



LIST OF ILLUSTRATIONS xix 

PAGE 

Plotting Whole Circle Bearings by Tangent Method . . 161 
Method of Checking an Unclosed Traverse . . . .162 
Checking Unclosed Traverse by Bearings to Lateral Object 163 
Graphical Adjustment of Clo.ung Error .... 164 

Distribution of Error in Traverse 165 

Latitudes and Departures of Survey Lines . . . .167 
Calculation of Latitudes and Departures from Whole Circle 
Bearings .......... 168 

Closing Error of a Polygon 178 

Linear Adjustment of Closing Error 178 

„ 179 

Plotting Survey Stations by Co-ordinates . . . . 180 

Construction of Bounding Rectangle . . . . .181 

Alteration of Bearings . . . . . . ..183 

Fundamental Problem in Triangulation . . . .186 

Closing Error in Triangulation 190 

Broken Base Line . . . . . . . .193 

Enlarging a Base Line ....... 193 

Series Method of Reading Angles . . . . .201 

Arrangement of Small Triangulation 202 

Polygons for Plotting Small Triangulation .... 203 
Perpendicular from a Point to a Line ..... 208 
Perpendicular from an Inaccessible Point .... 209 

Setting out a Parallel Line 209 

Running Straight Line between Two Points . . . 210 

...... 210 

Distance to Inaccessible Point 211 

211 

Distance between Inaccessible Points . .... 211 
Distance of Boat from Point on Shore -. . . . 212 

Length of a Line from Latitude and Departure . . .213 

Length of Line from Co-ordinates 213 

Ranging a Straight Line . . . . . . .214 

Survey Line Crossing a Sheet Boundary . .... 214 

Two Omitted Measurements in Traverse Polygon . .216 

• • 216 

Three-point Problem 218 

Height by Theodolite 219 

......... 220 

Curvature of the Earth 220 



xx LIST OF ILLUSTRATIONS 

FIG. PAGE 

189. Finding Height of Point above the Theodolite . . .221 

190. Finding Elevation of Point below the Theodolite . . . 222 

191. Mechanic's Water Level 226 

192. Axis of Bubble Tube 227 

193. Mechanic's Level 227 

194. Use of Mechanic's Level 228 

195. Dumpy Level 229 

196. Troughton and Simms Pattern Level 229 

197. Wye Level 229 

198. Diaphragm of Level 230 

199. „ „ 230 

200. „ „ 230 

201. Four-screw Levelling Arrangement 230 

202. Three-screw Levelling Arrangement 231 

203. Solid or Scotch Staff . . . . . . .234 

204. Section of Staff 234 

205. Portion of Staff seen in Telescope 235 

206. Alternative Graduation of Staff 236 

207. Sopwith Staff 236 

208. Bench Marks 239 

209. Levels by Rise and Fall Method 240 

210. Levels by Instrument Height Method 241 

211. Continuous Levelling . ..... 243 

212. Staff not held Vertical 253 

213. Error at Bottom of Staff 254 

214. Effect of Curvature and Refraction ..... 255 
Plate III. — Example of Section for Small Sewer Facing 262 
Plate IV. — Portion of Working Longitudinal Section of 

Railway Facing 266 

Plate V.— Portion of Working Plan of Railway. „ 268 

215. Plotting Cross -section 271 

216. Series of Cross-sections ........ 272 

217. Typical Contour Plan 273 

218. Section Plotted from Contour Plan 273 

219. Contour Plan of Volcanic Crater 275 

220. Section of Volcanic Crater 275 

221. „ „ „ 275 

222. Cross-sections for Contour Purposes 278 

223. Finding Positions of Contours ...... 279 



LIST OF ILLUSTRATIONS xxi 

PAGE 

Finding Positions of Contours 279 

280 

Use of Contour Plan in Designing 281 

Levelling up Steep Slope 285 

„ 285 

Levelling over Summit . . . . . . . 285 

Levelling across Hollow . . . . . . . 286 

Level of Overhead Point 286 

Levelling past High Wall 287 

Reciprocal Levelling ........ 288 

Setting out Curve by Radius . . . . . . 293 

Offsets to Curve from Tangent 293 

New Tangent in Setting out Curve 295 

Offsets at Equal Curve Intervals ..... 296 

Offsets from Chords produced . . . . . . 298 

Setting out Curve by Chords and Offsets .... 299 

Curve Problems . . 299 

300 

Setting out Curve by Theodolite . . . ... 302 

Changing Position of Theodolite 303 

Setting out Curve 305 

Curves to Right and Left ... . . . . 308 

Setting out a Bridge 311 

Area of Figure by Subdivision . . . . . .317 

Single Triangle equal to Quadrilateral . . . . . 317 

Area by Single Equivalent Triangle . . . . .318 

Area within Irregular Boundaries 319 

Area by Parallel Strips . . . . . . .320 

Areas by Offsets 321 

„...'" 321 

Areas by Mean Ordinates 322 

„ " „ „ ....... 322 

Area by Offsets . . . 322 

Elements of Planimeter ....:.. 323 

323 

Amsler Planimeter . . . . . . . .324 

324 

Area from Field Measurements . . . . . .326 

326 

326 



xxii LIST OF ILLUSTRATIONS 

FIG. PAGE 

264. Area of a Traverse 327 

265. Shrinkage of Plan 328 

266. Excavation in Foundation Pit 334 

267. Earthwork for Pond 336 

268. Earthwork in Levelling an Area ...... 337 

269. Railway Cutting and Embankment ..... 339 

270. Earthwork in Eailway Embankment ..... 340 

271. Adjustment of Level 344 



SURVEYING 

CHAPTER I 

SURVEYING — FUNDAMENTAL PRINCIPLES 

This chapter deals with the purpose of surveying and the simple 
geometrical principles, involving measurements of lengths and 
angles, on which it is based. The nature of the general problem, 
having regard to the fact that the earth's surface is not plane but 
curved, is treated briefly, and indication is given as to when the 
effect of curvature may be neglected. 



Use of Survey Plans. — For many purposes connected with the 
work of the land surveyor, architect and civil engineer, such as the 
calculation of areas of ground, recording of boundaries, designing 
and laying out of buildings, roads, railways and other works, it is 
necessary to have a plan of the portion of ground concerned showing 
in accurate proportion the principal features of the surface. 

Methods of Representing Surface of Ground on Reduced Scale. — If 

a portion of ground is not level, a complete representation of its 
surface on a small scale could only be given by means of a model 
constructed to show accurately the varying altitudes of the surface 
as well as the features that are evident in a horizontal projection. 
Such a model of a portion of ground is occasionally found useful for 
particular purposes. In ordinary work, however, it is the practice 
to represent the features of the ground by means of lines and con- 
ventional markings drawn on a flat surface, usually a sheet of paper. 
On such a surface it is possible to show directly only a horizontal 
projection of the features of the land, and this is what any ordinary 
survey plan shows. 

Plane Surveying. — Plane surveying denotes the methods and 
procedure which are applicable to surveying an area of small extent, 
where the earth's surface can be assumed to be a horizontal plane 

s. B 



2 SURVEYING 

and the effect of curvature is negligible. Plane surveying only will 
be dealt with in this book, but the effect of curvature will be 
investigated briefly and indication given as to when the error 
involved in neglecting it becomes practically appreciable. 

" Surveying " : What it Includes. — Surveying comprises all the 
operations necessary for the determining and recording on the plan 
of all the features of the surface of the ground, features of altitude, 
as well as those such as boundaries, streams, houses, &c, which are 
ordinarily shown in horizontal projection. It includes the taking 
of a connected series of measurements on the ground, so arranged 
as to fix the true relationship of the features of the ground to each 
other, the laying down of these measurements in correct proportion 
on paper, and the representing therefrom of the boundaries, streams, 
houses, &c, by means of lines drawn in ink. 

Under surveying we may also properly include the " setting out 
of works," as exactly the same principles are involved, but the 
operations are carried out in the reverse order. In this case the 
outlines and forms of intended works, such as buildings, roads, 
railways, &c, having been devised and drawn to scale on a plan, 
require to be marked off full-size on the ground to enable construc- 
tion to proceed. 

As survey plans are commonly used for the purpose of ascer- 
taining areas of land and quantities of earthwork in connection 
with estate, architectural and engineering work, it will be expedient 
to consider the principles of making the computations for the above 
purposes as falling directly under surveying. 

Surveying based on Simple Geometrical Principles. — The practice 
of surveying is based directly on very simple geometrical principles 
involving the measurements of lengths and angles, and, funda- 
mentally, has to deal with the fixing of the position of a point on 
the ground in relation to two other points whose relative positions 
are already known. The following four methods of doing this are 
in common use in surveying : — 

(1) By one linear measurement when the points lie in one straight 
line. 

(2) By two linear measurements. 

(3) By one linear and one angular measurement. 

(4) By two angular measurements. 






SURVEYING— FUNDAMENTAL PRINCIPLES 3 

Method by One Linear Measurement. — The first method is 
applicable where the point to be fixed lies on the line joining the two 
known points, or on the extension of that line. In this case a 
single measurement of length suffices to determine the point. Let 
A and B (Fig. 1) represent the two known points, and let C be a 
point lying on the line joining AB, and D a point lying on the 
extension of that line. Point C would be fixed with reference to 
A and B by measuring either the length AC or the length BC; 
similarly a measurement of the length BD fixes the point D. Points 



Fig. 1. 



C and D would be plotted on the paper by marking ofE to the proper 
scale, along the straight line drawn through the points A and B, 
the measured lengths of AC and BD. It is evident that the posi- 
tions of any number of points may be fixed along a straight line by 
this method, which is exemplified in the ordinary procedure of 
chaining a survey line, as described in Chapters III. and IV. 

Method by Two Linear Measurements. — Let A and B (Fig. 2) 
represent the two known points, 
and let C be the point whose 
position is required to be fixed 
with reference to points A and B, 
and assume also that all lie 
in one horizontal plane. To 
fix point C by method No. 2 
it is necessary to measure on the FlG - 2.— Fixing a Point. Method 
ground the straight horizontal 

lengths from A to C and from B to C. The distance from 
A to B is, of course, already known. To plot the points on 
paper, first mark off points A and B at the correct distance 
apart by scale. Then, with a pair of compasses, sweep intersecting 
arcs from A and B as centres with radii equal to the lengths of 
AC and BC respectively, to the scale of the plan. The point of 
intersection of the two arcs represents approximately the position 
of point C with reference to points A and B, the degree of approxi- 

B 2 




4 SURVEYING 

mation depending on the accuracy with which the various lengths 
have been measured and plotted. With perfect accuracy through- 
out, the triangle ABC on the paper would be an exact representation 
in miniature of the triangle formed by the three points on the 
ground. A point having been determined, as above described, by 
this method, it may then be used as a known point from which to 
fix others. Thus, in Fig. 2, point C having been fixed, the points 
A and C may be used as known points from which to fix another 
point, D, and so on successively. This is the principle on which 
the main points and lines are determined in Chain Surveying. 

Method by One Linear and One Angular Measurement. — To fix 
point C by method No. 3, as shown in Fig. 3, it is necessary to 
measure the straight horizontal length between the points A and C, 




D r 












""/€ 








/ \ 






/ 




\ 


i / 






\ 










'iXmeas ure 


angle 


****** 


'*%«feA 











Fig. 3.— Fixing a Point. Fig. 4.— Fixing a Point. 

Method No. 3. Method No. 4. 

and the horizontal angle which the direction of the straight line AC 
makes with the direction of the straight line AB. The similar 
measurements made from point B, namely, the length BC and the 
angle at B, would, of course, equally suffice to fix point C. To plot 
point C on paper a protractor may be used to lay off through 
point A a straight line making the observed angle with the line AB. 
The observed length of AC being then marked off to scale along this 
line, the point C is fixed. Other methods of plotting point C may, 
however, be used. For example, the length from B to C may be 
calculated by the principles of trigonometry, and point C may then 
be plotted as described under method No. 2. 

This method may be extended to the fixing successively of any 
number of points. Thus, a point D may be fixed with reference to 



SURVEYING— FUNDAMENTAL PRINCIPLES 5 

C and A by measuring the length CD and the angle which CD makes 
with CA. This is the principle on which Traverse Surveying is 
based. 

Method by Two Angular Measurements.— In method No. 4, illus- 
trated in Fig. 4, the measurements to be taken on the ground are 
the magnitudes of the horizontal angles which the directions AC 
and BC make with the direction AB. Point C could evidently 
then be plotted by repeating those angles on the paper, that is, 
by laying off with the aid of a protractor a straight line through 
point A making an angle with AB equal to the corresponding angle 
measured on the ground, and by laying off through B a straight line 
making with BA an angle equal to the observed angle at B. The 
intersection of these two straight lines would determine point C. 
In this case also, point C may be plotted by other methods, the 
usual procedure being to calculate the lengths AC and BC by the 
principles of trigonometry and then to plot the point by method 
No. 2. Method No. 4 can, like the other methods, be applied 
successively to the determination of any number of points. Thus, 
point D may be fixed with reference to A and C by observing the 
angles which the directions AD and CD make with the now known 
direction AC. This is the principle on which the main reference 
points are determined for most surveys of large extent. It is known 
as Triangulation. 

Methods applied to determining Altitudes of Points. — It might on 
first consideration seem that the principles involved in determining 
the altitudes of points above a fixed horizontal plane would neces- 
sarily be different from those involved in determining the relation- 
ship of points in a horizontal plane. The principles are, however, 
exactly the same, and are practically limited to methods Nos. 1, 
3 and 4 already described, although method No. 2 is of quite possible 
application. 

Method No. 1 applied to the determining of relative altitudes is 
represented by the process of ordinary levelling by means of a 
levelling instrument. In this case successive horizontal planes 
are given by the line of sight of the instrument when placed in 
different positions, and the vertical distances between the planes 
are deduced from the readings of the line of sight on a graduated 
rod or staff held vertically as described in Chapter XVI. 



6 SURVEYING 

Corresponding to method No. 3 for fixing points in a hori- 
zontal plane, we have the method, illustrated in Fig. 5, for deter- 
mining the vertical difference in height between two points. A and 
B are two points on uniformly sloping ground. The measurements 
required on the ground are the straight distance from A to B 
measured along the slope and the vertical angle made between the 
direction AB and a horizontal line AC in the same vertical plane 
as AB. The vertical height between the points can be arrived at 
by plotting the line AB in relation to a horizontal line AC, but in 
practice would usually be got by calculation. See Chapter XV. 

The determining of the altitude of a point by method No. 4 is 
illustrated in Fig. 186, Chapter XV. 

Effect of Curvature of Earth. — In what precedes, the surface of 
reference has been intentionally limited to a horizontal plane. As, 



Fig. 5. — Measurement of Altitude. 
Method No. 3. 

however, the surface of the earth, apart from the irregularities of 
hills, valleys, &c, is a curved surface, it would not be possible, to 
represent accurately on sheets of paper the features of any con- 
siderable area on the assumption that the surface was a plane. 
In dealing with any large extent of ground the surface of reference 
is taken as the ideal surface formed by the mean level of the sea 
assumed as continued and completed right round the globe. 

Fig. 6 represents a section taken through the centre of the earth 
and a portion of its surface, the curvature of the earth and height 
of the ground being enormously exaggerated. Point represents 
the centre of the earth, and A and B are two points on the surface 
of the ground whose distance apart it is required to find, referred 
to mean sea level. The line AO joining point A to the centre of the 
earth represents the vertical direction at point A. Similarly line 
BO represents the vertical direction at point B. It must be clearly 



SURVEYING— FUNDAMENTAL PRINCIPLES 



kept in mind that the vertical directions at two different points 
on the earth's surface, even when the points are quite close together, 
are never parallel. For two points which are one geographical 
mile apart the angular inclination between the verticals is one 
minute. For points 100 ft. apart the inclination of the verticals 
is roughly one second. In the figure point A' represents the pro- 
jection of point A at mean sea level, and B' is the projection of 
point B. The distance between the points A and B referred to 
mean sea level is then the length A' B' measured along the curved 
line marked mean sea level. It is evident that this length is less 
than the level distance B6 between the points measured at the level 
of B, and still less than the level 
distance k.a measured at the level 
of point A. The extent of the 
difference, however, is not great 
for moderate difference of altitude. 
Let the vertical height B'B be 
one mile and take it that the 
earth's semi-diameter is roughly 
4,000 miles, then the length B6 
would be greater than the length 
A'B' by ^nWth part. This extent 
of error might or might not be 
negligible, depending on the pur- 
pose of the survey, but in what 
follows it will be assumed that for 
ordinary purposes and moderate 
altitudes the lengths need not be reduced to correspond with mean 
sea level. 

As to neglecting Curvature of the Earth. — When a survey is of 
small extent also, the further assumption will be made that the 
surface of the earth is plane, no account being taken of its curvature. 
A survey may be considered as of small extent when its dimensions 
do not exceed ten miles by ten miles. A circular portion of the 
earth's surface ten miles in diameter has approximately the form 
of a circular curved watch glass, and is dished to the extent of 
17 ft. in the centre. The differences between any lengths or areas 
measured on such a surface and the corresponding lengths or 
areas measured on a plane surface ten miles in diameter are 




Curvature of Earth. 



SURVEYING 



almost inappreciable. For an area 100 miles in diameter, which 
is dished nearly 1,700 ft. in the centre, the curvature of the 
surface would require to be taken into account. 

Curvature of the Earth to be considered in Finding Altitudes. — The 
foregoing considerations refer only to the curvature of the earth 
so far as affecting the making of a plan showing a horizontal pro- 
jection of the surface features. In the operations required for the 
determining of altitudes the effect of curvature of the earth's surface 
becomes appreciable at very much smaller distances. Referring 
again to Fig. 6, suppose that a levelling instrument is set up at A 




Fig. 7.- 



-Measuring Horizontal Projections of 
Lengths and Angles. 



so as to give a line of sight Ad perpendicular to the vertical at A, 
that is, the line Ad is a horizontal line at the point A. Let d be the 
point where it strikes the vertical through B. Then dB does not 
give the true difference in height between the points A and B, but, 
due to the curvature of the earth, it is greater than the true height 
by the amount da. At a distance of one mile from A the distance 
da representing the deviation of the earth's surface due to curvature 
would be 8 ins. approximately. The deviation varies as the square 
of the distance from A, so that at one-eighth of a mile it would 
amount to | in. Therefore, the horizontal line of sight of a levelling 
instrument may only be taken as defining a level surface within 
about a radius of 200 yards if the error is limited to -| in. 



SURVEYING— FUNDAMENTAL PRINCIPLES 9 

Horizontal Projections of Lengths and Angles to be Measured. — In 

describing the methods of fixing the relationship of three points in 
plan, the assumption was made that all three points lay in one hori- 
zontal plane, on level ground. When the points do not lie in one 
horizontal plane the distance between the points measured along 
the surface of the ground will not be the proper distance for use in 
plotting a horizontal projection of the points. Let A and B (Fig. 7) 
be two points on uneven ground, and let C be the elevation of a 
third point on a hill some distance beyond the vertical plane through 
A and B. Let A', B', C, be the projections of the three points on a 
horizontal plane. Then in measuring from A to B the distance to 
be determined is not the length along the ground surface ADB, 
nor yet the direct distance AB, but the horizontal projection of 
the latter, that is, length ab or A'B'. Similarly in measuring 
angles. What is required is not the actual angle between two lines 
in a plane containing the lines, but the horizontal angle made by 
the projections of the lines on a horizontal plane. Thus in measuring 
for surveying purposes the angle between the lines joining point C 
to points A and B we desire to find, not the angle made by these 
lines in the sloping plane containing them, but the angle A'C'B' 
made by the projections of these lines on a horizontal plane. 

Difficulties of Surveying not in Principles but in their Application. — 
From the descriptions given of the methods of fixing the horizontal 
and vertical positions of points when the surface of reference is a 
horizontal plane, it will be recognised that the principles of sur- 
veying are exceedingly simple. The difficulties of surveying lie not 
in its fundamental principles, so long as the extent of surface to be 
surveyed may be considered as practically plane, but in the applica- 
tion of them in such a manner as will best suit the character of the 
ground and give the necessary degree of accuracy with economy of 
time and labour. No measurement of length or angle taken on the 
ground can be considered as absolutely exact. There is always 
a certain amount of error, and similarly no measurement taken 
on the ground can be reproduced in absolutely exact proportion on 
the paper. A plan, therefore, only represents the features of the 
ground with a certain degree of accuracy. A very important part 
of the practice of surveying has to deal with the methods and 
precautions to be adopted in order to reduce mistakes of measure- 
ment and plotting to the smallest possible amount. 



CHAPTER II 

CHAIN SURVEYING— INSTRUMENTS 

Chain surveying is of very limited application. It is quite 
suitable for surveying small areas on easy ground, and might on 
occasion be employed for larger areas if instruments for measuring 
angles were lacking. It is not a practicable method for large areas 
or uneven ground. It has been described somewhat fully in the 
following pages, because much of the detail work of chain surveying 
is equally applicable to other methods of surveying. This chapter 
deals with the linear measures, British and metrical, usually em- 
ployed, and with the instruments used in chain surveying — chains, 
steel bands, tapes, offset staff, measuring rods, arrows, ranging 
poles, &c, cross staff, optical square, optical prisms, line ranger, 
inclinometer, Abney's level, the human eye. 



British Units of Linear Measurement. — The British standard for 
measures of length is the Yard. The following sub-divisions and 
multiples of the yard are used for convenience in stating the measure- 
ments of lengths of widely varying magnitude : — 
The Inch = ^ P ar t of a yard or T \ part of a foot. Sub-divisions 

of the inch are expressed either decimally or by the fractional 

parts \, \, \, &c. 
The Foot = | part of a yard. This is the unit of measurement 

principally used for surveying purposes in Britain and its 

Colonies, and in the United States of America. 
The Fathom = 2 yards or 6 ft. This unit is often used in figuring 

the depths of soundings on nautical charts, and in connection 

with mining for stating the depths of shafts, &c. 
Gunter's Chain = 22 yards or 66 ft. Used to some extent for 

surveying purposes in Britain, and useful for obtaining 

areas in acres. 
Furlong = 220 yards = 10 chains. 
Statute Mile = 8 furlongs = 1,760 yards = 5,280 ft. 



CHAIN SURVEYING 11 

Metrical Units. — Throughout the principal European countries 
(except Russia, which has the British foot as the basis of measure- 
ment), and also South America, the unit of linear measurement 
is the Metre. 

One Metre = 39-37043 ins. = 3-28087 ft. 
= approximately 3 ft. 3f ins. 
The sub-divisions of the metre are : — ■ 
Decimetre = xu metre. 
Centimetre = t^q metre. 
Millimetre = xoVs metre. 
The multiples are : — 

Decametre =10 metres. 
Hectometre = 100 metres. 
Kilometre = 1000 metres. 
The units most commonly used are the kilometre, metre, centi- 
metre and milhmetre. 

The following approximate rules are very useful for mentally 
changing British units into metrical units and vice versa :— 

To change metres into feet divide 10 times the number of metres 
by 3. Thus 36 metres = ^p = 120 ft. approximately. To 
change feet into metres multiply by 3 and remove decimal point 
one place to the left. 

To change centimetres into inches multiply by 4 and remove 
decimal point one place to left. Thus 15 cm. = 1-5x4=6 ins. 
Similarly, to transfer inches into centimetres, divide 10 times the 
number of inches by 4. Thus 12 ins. = -f° = 30 cm. approx. 

The values determined by the above methods are wrong to the 
extent of fully 1^ per cent. 

Measuring Instruments. — The instruments ordinarily used in this 
country for measuring continuous long lengths on the ground are 
the " Chain " and the " Steel Band." 

Chain.— Chains are made either 66 ft. or 100 ft. long. They are 
formed of straight links of steel wire joined to each other by three 
small circular or oval wire rings. The 66-ft. chain, known as 
Gunter's chain, is divided into 100 links each -66 ft. long. The 
divisions are marked by the middle small ring at each joining. 
The 100-ft. chain is divided into feet. Every tenth link or foot 
is marked by a brass tag shaped so as to indicate its position at a 



12 SURVEYING 

glance. Every intermediate fifth division is also marked. See 
Figs. 8 and 9. 

Steel Band. — The steel band consists of a thin flexible strip of 
tempered steel, having a cross section of about § in. X 50- in., or 
narrower, provided with a brass swivelling handle at each end, and 
marked off into feet or links. A very narrow band is less apt to 
break by kinking. Fig. 10 illustrates the method of marking the 
main divisions. The tenth, twentieth, thirtieth, and fortieth 
divisions from each end are marked by oval brass plates fixed on 
each side of the band and having one, two, three, and four round 
spots respectively. The centre or fiftieth division is marked by a 
blank oval plate on each side. The intermediate fifth divisions 
are marked by large-headed brass rivets, and the remaining divisions 
by small-headed rivets. The foot or link at each end of the band 
is usually divided decimally. See Fig. 11. 

Etched Steel Band. — Steel bands can be obtained fully divided 
along the whole length into feet and inches with the divisions shown 
by etching. Such bands are useful where intermediate distances 
have to be read very exactly, but they are difficult to read and keep 
clean. 

Steel Tape. — For exact detail measurements a steel tape is used. 
This consists of a light strip of steel marked as described above for 
the etched steel band. Usual lengths are 50 ft., 66 ft., and 100 ft. 
The free end of the tape has a brass finger loop, and the outside of 
this loop is the zero of graduation. 

Linen Tape. — For the detail measurements required in ordinary 
surveying a linen tape is generally used. This consists of a strip 
of woven linen tape about f in. wide, well varnished, and with the 
divisions and figures painted on. Feet and inches are usually shown , 
on one side of the tape and links on the other. In many tapes the 
feet are marked in red figures and the inches in black figures, as 
shown in Fig. 13. For use in surveying, a simpler method, as shown 
in Fig. 12, is preferable. Linen tapes are liable to considerable 
extension in ordinary use and to shrinkage when wet. Tapes 
with interwoven metallic threads are of more constant length. 

Offset Staff. — An " offset staff " consisting of a graduated wooden 
rod, which may be 10 ft. or 10 links long, is useful for measuring 






CHAIN SURVEYING— INSTRUMENTS 



13 



5Q 



*s§? 



<S§§ 



1 

00 




05 

I 



14 SURVEYING 

short distances. It can be used by one person, whereas a tape 
requires two persons. 

Wooden Measuring Rods. — Graduated wooden measuring rods, 
shod with metal at each end, and finished very exactly to a definite 
length, are sometimes used in Britain for special purposes, such as 
setting out steelwork. For measuring any considerable length 
two rods at least are used. They are placed end to end successively 
in the manner in which a joiner uses two 3-ft. rules to measure an 
exact distance. In Germany wooden rods 5 metres long are com- 
monly used for measuring survey lines where great accuracy is 
required, especially in town surveying. The accuracy is said to 
be considerably greater than that attainable by the use of steel 
bands. 

Metre Chain, &c. — The metre chain or band used in surveying 
has generally a length of 20 metres. 

Arrows. — A record of the number of whole chain lengths measured 
is kept by means of a set of arrows, usually ten. These are formed 
of stiff steel wire, pointed at the lower end for fixing into the ground 
and having a ring at the top for carrying purposes, the overall 
length being usually 12 ins. The arrows are rendered conspicuous 
by having a piece of bright red cloth attached to the ring. 

Ranging Poles. — The main survey points on the ground and, 
where necessary, intermediate points on the straight lines joining 
them are marked by means of wooden poles. These have usually 
a length of 6 ft., including the point, but may vary from 5 ft. to 
10 ft. The poles are usually octagonal in cross-section and taper 
from bottom to top. They are shod at the bottom with a heavy 
iron point, and are painted in alternate lengths either black and 
white, or red and white, or black, red and white in succession. 
In order that the poles may, if occasion requires, be used as offset 
staffs, they should be painted in lengths of 1 ft. for a survey con- 
ducted with a 100-ft. chain, and in link-lengths for a survey made 
with a 66-ft. chain. Poles are sometimes used of round cross- 
section, but these are not so convenient for ranging out straight 
lines by the eye as octagonal poles. The gradual shading of light 
that occurs round a cylindrical body causes the edge of a round pole 
to appear very indefinite under certain views. 



CHAIN SURVEYING— INSTRUMENTS 15 

Flags. — To enable a pole at a distance to be more easily picked 
out, a small triangular white or red flag may be attached to its 
top. 

Laths and Whites. — Intermediate points on a line in open level 
ground may be very conveniently lined out with straight laths about 
2 or 3 ft. long. They are also useful for marking points which 
have been lined out by poles, if the poles require to be removed 
for use elsewhere. Small straight twigs or shoots cleft at the top 
and holding a piece of white paper are useful for the same purpose. 

Pegs, &c. — The principal survey points require to be marked 
in such a manner that they may be readily found at any time 
during the progress of the survey. 
In fields and soft ground stout 
wooden pegs are generally used. 
Where survey points require to 
be marked on hard roads, cause- 
way, &c, pointed iron bolts, dog 
spikes, or large nails are used, 
driven in flush with the surface. 
In towns marks may be cut with 
a chisel on kerbs, pavements, &c. 

Cross Staff. — This instrument 

for setting off right angles, one 

form of which is illustrated in Fig. 

14, consists essentially of a frame 

, L ■-, , p Fig. 14.— Cross Staff. 

or a box mounted on top of a 

wooden rod, which is shod and pointed for fixing into the 

ground, the frame or box having four, sometimes eight, vertical 

slits arranged in its circumference, so as to give two fines of 

sight at right angles to each other. To set out a right angle to 

the chain line at a given point with this instrument, stick the rod 

in the ground at the point, turn one of the fines of sight till its 

direction coincides with the direction of the chain line as defined 

by ranging rods, then the other line of sight is now at right angles 

to the chain line, and, by viewing through the slits, a pole may be 

placed or a mark made at the point to which the offset should be 

taken. The use of the instrument to find where a perpendicular 

to the chain line dropped from an object, such as the corner of a 




16 



SURVEYING 



house, will strike the chain, is not so simple, as the process is one 
of trial and error. To get accurate results with this instrument 
care must be taken to see that it is set up as nearly vertical as 
possible. When the vertical slits are made fairly long in proportion 
to the diameter of the instrument, perpendiculars may be con- 
veniently set out on ground having a considerable slope. 

Optical Square. — A more handy instrument than the cross staff 
and one much more used nowadays is the optical square. This 
is a reflecting instrument, and serves to give by means of one sight 
the result attained by two separate sights with the cross staff. 
Fig. 15 illustrates the optical theory of the instrument. SU and 
TU represent horizontal sections through 
two vertical reflecting surfaces, and ES 
represents the direction of a ray of light 
striking the mirror SU at S and making 
an angle C with the normal to the 
mirror. By the laws of reflection this 
ray will be reflected in a direction 
ST such that ST also makes an angle 
C with the normal. Let ST strike 
the mirror TU at an angle d with the 
normal, then the reflected ray TA will also 
make an angle d with the normal. It is 
required to find the relation existing 
between the angle a which the mirrors 
and the angle b which the entering ray ES 







Fig. 15.— Principle of 
Optical Square. 



make with each other, 
makes with the departing ray TA 
In the triangle STU we have : — 

a + (90° — c) + (90° — d) = 180°, 
which gives : a = c + d. 

From the triangle BST we have : — 
b = 2c + 2d 
.-. b = 2a 
Therefore the angle between the entering and departing rays is 
equal to twice the angle between the mirrors, so that if the angle 
a be made 45° the angle b will be 90°. An eye then placed at E, 
and looking just over the top of the mirror SU towards a distant 
object C, would see in the mirror, apparently in the same vertical 



CHAIN SURVEYING— INSTRUMENTS 



17 



line with C, the image of an object A if so placed that the line BA 
is perpendicular to line BC. This is the principle of the arrange- 




Optical Square. 



ment and use of the optical square. Fig. 16 shows an external 
view of the instrument open, and Fig. 17 shows a plan, the lid being 




Section of Optical Square. 



removed. The lower portion of the case contains the two mirrors 
and has a vertical rim in which are formed the eyehole A, the 
aperture C for viewing the forward object, and the hole B for 
viewing the side object. The lid or top portion of the case is similar 
to the lower portion, over which it slips. When the instrument is 
s. c 



18 SURVEYING 

not in use all the openings are closed by rotating the lid. In Fig. 17 
the mirror F is firmly fixed to the bottom of the case. The mirror 
E is attached to the circular base shown, and this is connected to 
the case by a centre pin, about which it can rotate. The rotation 
of the mirror, required for purposes of adjustment, is effected by 
means of the screw D working in an arm attached to the base of 
the mirror as shown. The adjustment of the 
verticality of the mirror is effected by means 
of two screws whose heads project through the 
, bottom of the case. A key G, which screws 
into the lid of the box when not in use, is pro- 
vided for the adjustment of the mirror E. The 
instrument complete is about 2| ins. diameter 
and | in. thick. 



<9 



Use of Optical Square. — Fig. 18 illustrates the 

use of the optical square. Let a straight line 

ON be defined by poles erected at points 0, P, 

F J- G - 18 -— Use of M and N. It is required to find at what point 

on this line a perpendicular from a point L will 

fall. Holding the instrument to the eye and looking directly 

towards the pole at M walk forward along the straight line until 

the doubly reflected image of the pole or object at L is seen 

immediately under the pole at M. The optical square is then at 

the point required. The instrument is also used for setting out 

a perpendicular to a straight line from a ^ — . 

given point. The surveyor in this case O — L 

stands at the given point and sends an " Q- — ^. 

assistant with a pole to the required dis- ^^— ^ 

tance in a direction as nearlv at right angles FlG - 1 5"T St ?^ d for Use 

, . L , TT - y . ,° 6 . of Optical Square, 

as can be estimated. He then directs his 

assistant backwards or forwards until the reflected image of his 
pole appears in the same vertical line with the forward pole. 
When the perpendicular has to be set out to the right hand the 
instrument must be held in the left hand. To set out a perpen- 
dicular to the left hand the instrument must be held other side up 
in the right hand. It is better to use the same eye in all cases, 
say the right eye. The surveyor should experiment for himself 
with a plumb-bob to find out in what position he must stand so 



CHAIN SURVEYING— INSTRUMENTS 



19 



that the optical square may be held directly above the required 
point on the chain line. Fig. 19 shows the position the surveyor 
should take up in order to set off a right angle at the 50 mark on 
the chain, using the instrument at the right eye, the point on the 
chain being in line with the toes. 

Use of Optical Square on Sloping Ground.— The general usefulness 
of the optical square depends on the fact, not always recognised, 
that it can be used for accurately setting out a right angle on 
ground of any steepness provided it is possible to obtain a hori- 
zontal line of sight in the other direction. This use of the optical 
square is illustrated in Fig. 20. The surveyor standing in the field 
at A desires to fix a pole at C on the railway embankment in a line 
at right angles to the chain line denned by the poles at B and D. 




Fig. 20. — Setting out a Eight Angle on Sloping Ground. 

Let the horizontal line of sight from the eye at A strike the first 
pole at point B. Then the line AC and its horizontal projection 
on the ground will be truly at right angles to the line ABD, when 
on tilting the instrument so as to bring the image of the vertical 
pole at C into view simultaneously with the pole at B, any point 
of the pole C can be brought into coincidence with the point B. 
The angle BAC is then a right angle set out in a sloping plane, the 
side BA being, however, horizontal, and the projection of AC on 
the ground is also at right angles to AB. This may be shown by 
placing a right-angle set-square in a sloping position with one base 
resting on a horizontal table. 

Testing the Optical Square. — The best method of procedure in 
testing and correcting the optical square is as follows : On fairly 
uniform ground set out three poles A, B, C, in a straight line at, 
say, 50 yards apart, as shown in Fig. 21. With the instrument at 

c2 



20 



SURVEYING 



B sight towards C and set out the angle CBd v supposed to be a 
right angle, and fix a pole at d v Then turn round and, with the 
instrument turned upside down, sight towards 
A, set out the same angle XBd 2 , and fix a 
pole at d 2 , the distance Bd 2 being equal to 
Bdj and, say, about 50 yards. If d 2 coin- 



d 2 adjustment, the angle set out each time 

being 90°. If, however, d 2 does not coin- 
cide with d v fix a pole at D midway 
between the two points. Then the angle 
CBD will be a right angle, and with the 
Fig. 21. — Testing instrument at B the key must be used to 
Optical Square. tum t j ie mova ble mirror until the image of 

the pole at D is seen to coincide with the pole at C. Then test 
further by sighting towards A. If the image of D now also coin- 
cides with the pole at A the instrument is in correct adjustment. 

Prism Square. — The most reliable hand instrument for setting 
out right-angle offsets, &c, is the prism square. The principle of 




Fig. 22. — Principle of Prism 
Square. 



Fig. 23. — Principle of Five- 
sided Prism. 



one form of the instrument is shown in Fig. 22. The prism consists 
of a small block of glass, the upper and lower parallel surfaces 
having the form of right-angled isosceles triangles. All the faces 



CHAIN SURVEYING— INSTRUMENTS 21 

are ground and polished to make the exact required angles with 
each other, and the hypotenuse edge is silvered. A ray of light 
entering side AB in an oblique direction near the acute corner A, 
as shown in the diagram, and making an angle a with the normal 
to the face, is refracted and enters the glass, making an angle /3 
with the normal such that sin a — U sin /3 where U is a coefficient 
depending on the kind of glass. The ray then follows the path 
shown on Fig. 22, and a study of the diagram shows that the 
emerging ray is at right angles to the entering ray. 

The instrument is used in a similar manner to the optical square. 
A forward object is sighted directly over or under the prism, and 
simultaneously the image of a 
side object is seen in the prism 
after two refractions and two 
reflections. The right angle is 
given when the image of the side 
object appears in coincidence with 
the forward object. 

Fig. 23 shows the principle of 
another form of the prism, and i^^'^^ B^B --~ 
Fig. 24 shows a view of the 
corresponding instrument com- 
plete. As compared with the 

usual reflecting form of optical 

., • i_ ,i i Fig. 24. — Zeiss Prism Square, 

square, the prism has the advan- ^ 

tage of greater clearness. It can be used almost as long as 

objects are distinctly visible, while the optical square fails long 

before dusk. The prism has the further great advantage that, 

if it is correct to begin with, it remains correct so long as it is at 

all serviceable. 

Wooden Set-square. — For taking accurate small offsets, as required 
in town surveying, a large triangular wooden set-square is some- 
times used. Such a set-square, with the perpendicular sides, say, 
6 ft. and 4 ft. long, is also useful in setting out small details of 
buildings, bridges and similar work. 

Line Ranger. — This is a hand instrument used for obtaining with 
a close degree of approximation an intermediate point on the line 
between two survey stations without sighting from one end of the 




22 



SURVEYING 



line. The principle of a reflecting form of the instrument is shown 
in Fig. 25. Two small mirrors are fixed exactly at right angles to 
one another, one on top of the other, and mounted in a case, or 
simply on a handle. An examination of the diagram shows that 
if an object A is seen in the upper of the two mirrors immediately 
over their crossing point, then an object B will be seen in the lower 
mirror directly under the image of A, provided B, the instrument 




Fig. 25. — Principle of the Line Banger. 

and A are in one straight line. In using the instrument, therefore, 
the observer, having placed himself approximately in line, holds 
the instrument in front of his eye, and turns it so as to bring the 
image of A to about the centre of the upper mirror. Looking in 
the lower mirror he will see the image of B, but, unless by good 
luck, it will not appear directly under the image of A. He then 
moves backwards or forwards at right angles to the direction AB 
until the images of A and B appear exactly in the same vertical 




Fig. 26. — Inclinometer. 

line. The instrument is then in line between A and B and its 
position may be marked by a pole. Another type of the instru- 
ment consists of two prisms of the form shown in Fig. 22, 
mounted one on top of the other with their long sides at right angles 
to each other. It is used in the same manner as the reflecting type. 

Inclinometer. — In order to determine the true horizontal distance 
between two points on sloping ground when the measurement 
has been made on the slope, it is necessary to know the vertical 






CHAIN SURVEYING— INSTRUMENTS 



23 



angle of inclination of the ground. A simple instrument suited 
for this purpose is shown in Fig. 26. In using the instrument as 
illustrated in Fig. 27, a pole or staff is laid on the ground, and on it 
the lower bar of the instrument 
is placed. The upper bar is then 
tilted up to the horizontal posi- 
tion, as shown by the bubble of 
the spirit level coming to the 
centre of its run. The inclina- 
tion of the ground is then read off 
on the graduated arc to about the 
nearest half degree. The instru- 
ment may be used in a similar 
manner for rapidly taking rough cross-sections on steep ground. 

Abney's Level. — Abney's level is a useful hand instrument for 
measuring vertical angles, and is based on a very neat device. 
The principle of the instrument is shown in Fig. 28. A square 




Fig. 27. — Use of Inclinometer. 




Fig. 28. — Principle of Abney's Level. 

tube about 4| ins. long has a small eye-hole at one end. Near the 
other end a mirror inclined at 45° to the axis of the tube extends 
across half its width, the other half of the tube being open. A 
horizontal scratch extends across the centre of the mirror and the 
same line is produced across the open half of the tube by a fine 



24 SURVEYING 

hair or wire. Attached to the tube is a graduated arc of a circle 
about 3 ins. diameter, whose centre lies on a line drawn from the 
centre of the mirror at right angles to the axis of the tube. An 
index arm is hinged to the centre of the arc. A bubble tube is 
attached at right angles to the index arm in such a position that 
when the bubble is at the centre of its run it is also at the centre 
of the arc. An opening in the top of the tube allows the bubble 
to be seen in the mirror. An inspection of the diagram shows that 
when the bubble is seen bisected by the scratch in the mirror, 
it is at the centre of its run and the index arm is vertical. The 
angle a which the index reads on the arc is then equal to the angle 
a which the line of sight makes with the horizontal. Fig. 29 
shows the field of view in the tube when a sight is taken to the top 
of a pole. The hatched portion indicates the mirror with the image 
of the bubble bisected by the scratch. The unhatched portion 
shows the view through the open half of the tube. 

In using Abney's level to find the inclination of a line on the ground 
it is necessary to sight to an object at the same height above the 
ground as the observer's eye. The top of a pole of proper height 
forms a convenient object, but it will often be found expedient to 
observe simply to the head of an assistant. 

The Human Eye. — The human eye is an optical instrument 
which every surveyor on occasion uses instead of level, plumb-line 
or square. Unless where extreme accuracy is required, he may 
use it to tell him when the slope of the ground is such that its effect 
may be neglected in measuring along a survey line. In surveying 
details and taking offsets he uses it continually to tell him when the 
tape is held horizontally. He may use it to tell him whether a 
ranging pole or level staff is standing vertically, or whether one 
direction is at right angles to another. On fairly level and uniform 
ground, or where there are known horizontal and vertical lines to 
refer to, the normal eye will in the above cases give good approxi- 
mate results. On the other hand, in presence of certain configura- 
tions of ground, and under certain influences, it will give indications 
which are not only unreliable, but are absolutely misleading. A 
beginner, levelling on steep ground and working up or downhill, 
soon finds that in setting up his instrument he may easily misjudge 
the level of the bottom or top of the staff by several feet at quite 



CHAIN SURVEYING— INSTRUMENTS 25 

a short distance away. If he has much levelling to do along roads 
in undulating country he may happen on a stretch where his eye 
tells him definitely he is going downhill while his level proves he 
is going uphill. When walking along a straight railway embank- 
ment on steep sidelong ground he may notice that the permanent 
way appears canted up towards the deep side of the embankment 
while as a matter of fact the rails are level across. With no known 
vertical or horizontal object to guide him let him erect a ranging pole 
on steeply sloping ground, so as to be as nearly vertical as he can 
judge by the eye. He will find on testing with the plumb-bob that 
the pole is hanging considerably downhill. In the above-mentioned 
cases the configuration of the ground acts to produce an optical 
illusion. Due to a different influence, centrifugal force, is the 
familiar illusion whereby to a passenger in a train travelling round 
a curve external objects appear off the perpendicular. Enough has 
perhaps been said to show that the eye is not an independent instru- 
ment and, therefore, should not be used for estimating verticals, 
horizontals, &c, where accuracy is required. 



CHAPTER III 

CHAIN SURVEYING — FIELD OPERATIONS 

In this chapter consideration is given to the methods of ranging 
out straight lines by the eye on level ground, and over rises and 
hollows, as would be required in setting out the base-lines for a 
chain survey. The measurement of a distance with the chain is 
also described in detail, and attention is drawn to the more frequent 
sources of error in reading the chain. The methods of obtaining 
the true horizontal distance on sloping ground are fully gone into. 



Survey Party and Equipment. — The surveyor will have two 
assistants who will handle the chain and other instruments, while 
the surveyor himself directs the operations and records the measure- 
ments in the survey-book. The party will be equipped with the 
following instruments : — 

Chain or steel band (66 ft. or 100 ft.). 

50 ft. or 66 ft. linen tape. 

Set of ten arrows. 

Set of ranging poles, say six. 

Optical square or prism. 

Inclinometer. 

Plumb-bob. 

Ranging Lines. — The end points or stations of the survey line 
having been fixed, it is necessary, if the line is of considerable length, 
to establish and mark by poles, laths, or otherwise, some inter- 
mediate points for the guidance of the chainmen. Suppose, in the 
first instance, that the stations are fully 1,000 ft. apart, with fairly 
uniform and unobstructed ground between, and that it is desired 
to mark three intermediate points. Ranging rods are first set up 
at the end points and their verticality assured by testing with the 
plumb-bob. The surveyor remains at one station and directs an 
assistant to take with him the number of poles required for the 



CHAIN SURVEYING— FIELD OPERATIONS 27 

intermediate points and to proceed first to the most remote point. 
He will there, having guided himself as nearly into line as he can 
judge, stand facing the surveyor, and hold a pole erect and out to 
one side of himself so that the surveyor may have a clear view of it. 
He will then move in the direction signalled to him by the surveyor, 
a motion of the surveyor's right hand meaning that he is to move 
laterally towards the surveyor's right hand. A good method when 
the assistant is standing wide of the line is for the surveyor to keep 
his arm held out in the direction in which the assistant has to move. 
The latter will then keep moving in that direction until he sees 
the arm drop, when he will know he is nearly in line. He will then 
hold the pole erect, point on ground, and move it a little at a time 
according to the signals until it is in correct position. The signal 
for this is usually given by the surveyor raising both arms and 
bringing them rapidly down together, the gesture indicating that 
the pole is to be " planted " or stuck into the ground. On receiving 
this signal the assistant will press his pole a little distance into the 
ground, plumb it carefully, and then step aside to let the surveyor 
get another view. He will then shift it a little or press it home, 
according to the signal he receives. He must take particular care 
to leave it perfectly plumb. He will then proceed to the second 
point, and having now two poles behind him, he should tarn round 
and rapidly put his pole into line with these two. Then, on getting 
the signal from the surveyor, he will take a very small shift and 
rapidly arrive at the correct position. He will proceed in the same 
manner for the other points. 

The assistant who has to set poles in a survey line should pay 
attention to the following points : Before setting out to give the 
points he should, whenever possible, take a view along the line 
from one end station and take note of any mark, such as a tuft of 
grass, weed, stone, &c, occurring near the line at the position of 
his first point. He will thereby be enabled to hold the pole nearly 
on line at the first attempt, thus saving time. When carrying 
several poles with him, he must be very careful not to hold a pole 
out for lining with one hand while at the same time retaining the 
spare one or two in the other hand, and in view of the surveyor, 
as much confusion and annoyance may thereby be caused. When 
he is holding out a pole for lining, the spare ones should be lying 
on the ground. 



28 SURVEYING 

It cannot be too much emphasised that good lining out can only 
be accomplished when the poles are held and planted truly vertical. 
The assistant must, therefore, train himself to hold the poles cor- 
rectly and to test them rapidly. Rough-and-ready approximations 
to the vertical may be got, in the absence of a plumb-bob, by 
holding a pole lightly at the top between the thumb and a finger, 
and letting it hang free, or by dropping a small stone. It is better, 
however, to improvise a plumb-bob by attaching any small heavy 
object to the end of a string. 

The usual method adopted by the person who is directing the 
poles into line is as follows : Standing or crouching behind the 
station pole he puts one eye in line with the outside edges on one 
side of the two station poles and signals the assistant with the third 
pole until its edge appears in the same line. He tests by looking 
along the edges on the other side. To get good results he must not 
sight from close beside the station pole, but from some distance 
behind it. The following is a more rapid and satisfactory method 
than the above : The surveyor stands with both eyes open a few 
yards behind the station pole and looks fixedly at the distant pole. 
He sees two images of the near pole, as it is out of focus. He places 
himself so that the distant pole appears central between these two 
images and then directs the intermediate pole into position, so 
as to completely cover the distant pole. It is then in correct 
line. 

In practice, survey lines may often be ranged by the method of 
production. A pole is fixed at one end of the line and another is 
placed a short distance ahead, say, 200 or 300 ft., in the direction 
in which the fine is to run. A third pole is then fixed a similar 
distance ahead by sighting back on the two already planted, and 
so placing it that all three appear in line. The fine is extended to 
the required length by fixing additional poles successively in the 
same manner. On uniform ground straight lines 2,000 ft. long may 
be run very accurately by this method. 

When the purpose of ranging a straight line is merely to enable 
the distance between the end stations to be measured, little refine- 
ment is required in fixing the intermediate points. A deviation 
of 1 ft. in a fine several hundred feet long would give rise to no 
appreciable error in the length of the line. If offsets were taken 
from the line, however, they would be in error to the extent of the 



CHAIN SURVEYING— FIELD OPERATIONS 29 

deviation. Particular care must, therefore, be exercised in the 
ranging of lines from which offsets are to be taken. 

Ranging Line across a Hollow. — When a depression intervenes 
between the end stations, so that poles held at intermediate points 
come wholly below the direct line of sight between the stations, 
or when a hill occurs to prevent the one station from being visible 
from the other, the ranging of an accurate straight line becomes a 
difficult matter. On ground which gives rise to many such cases 
good results cannot be expected from chain surveying. The usual 
method of ranging out a line across a hollow between two points A 
and B (Fig. 30) is as follows : The surveyor standing at A directs 
an assistant to a point C, such that the top of a pole held there is 
at the level of the line of sight from A to the bottom of pole at B. 





WlllllllllllDL 
Era. 30. — Banging Line across a Hollow. 

He then signals the pole at C into line, which must be planted 
perfectly vertical, as otherwise the bottom might be out of line. 
A pole may then be lined out to a point D further downhill, by 
sighting to the bottom of the pole at C. Successive points may be 
put in line in this manner until perhaps a point in the hollow is 
reached, which is invisible from A. 

A more expeditious method of proceeding after the pole at C is 
fixed is for the assistant there, sighting to the bottom of pole A, 
to line out a pole held by the surveyor at E. The surveyor at 
E will then line out his assistant at D, and so on, each moving down- 
hill alternately. 

A method which is often useful, and is generally appropriate 
when the valley to be crossed is fairly deep, is to hang a plumb-bob 
over the station A (Fig. 30) by the method shown in Fig. 31. The 
surveyor then stands as far back from the plumb-line as he can 



30 



SURVEYING 



without limiting his view into the valley, puts his eye into range 
with the pole at B and the plumb-line, and then sights down the 
plumb-line and signals the pole at C or D into line. He must 
practically sight to B and C simultaneously. 

Ranging Line over a Hill. — The method of ranging a straight 
line between two points A and B over an intervening hill is illus- 
trated in Figs. 32 and 33. 
The surveyor goes to a 
point C on one side of the 
crown of the hill, where he 
can j ust see the top of the 
pole at B, and as nearly in 
the straight line between 
A and B as he can esti- 

wiiiiiiiiiiiiiiiii. mate - He directs ^ 1S 
* assistant to go down the 

Fig. 31.-Device in Banging Line. other side q£ ^ m ^^ 

he can just see the top of the pole at A, and then to put his pole 
into range with A and C. The surveyor standing at C now sees 
that the assistant's pole at D is not in line between poles C and B. 
He, therefore, directs the assistant to move laterally to point D' 





Figs. 32 and 33. — Banging Line over a Hill. 

in line between C and B. The assistant at D' now directs the sur- 
veyor to C in line between D' and A. They proceed in this manner, 
each putting the other in line alternately, until it is found that 
the intermediate poles range simultaneously with the extreme poles. 
It need hardly be pointed out that the above procedure, which 






CHAIN SURVEYING— FIELD OPERATIONS 



31 



necessarily involves sighting to 
the upper ends of poles, is not 
conducive to accurate work, and 
that, if more than one undula- 
tion occurs between the stations, 
it may be quite impracticable 
to run the straight line without 
the use of the theodolite. 

Ranging past Obstacles. — 

Objects such as trees, hedges, 
&c, occurring on a survey line 
may often form as great hind- 
rances to the ranging of a line 
as hills or valleys. Some devices 
for ranging past or over obstacles 
are described in Chapter VII. 
As no two cases happen exactly 
alike, the surveyor must hold 
himself ready to invent or adapt 
a device to suit each circum- 
stance. 

Chaining Survey Lines. — A 

new steel band should, before 
using, be tested on an official 
standard at ordinary tempera- 
ture and its error, if any, noted. 
It will not require further testing 
unless after repairing, due to 
breakage, or if the handle 
swivels should become worn. 
The official standard, as laid 
down in some of the large towns, 
for testing 100-ft. chains is illus- 
trated in Fig. 34. At one end 
of a heavy, continuous horizontal 
granite base a block of brass 
with a vertical face is fixed. 
This face is the zero of the 



m 








(0 
uJ 

I 

o 
















O 

<* 

m 




(/ill. 
Id 

I 
o 

z 























II 
El 

EB 
El 
El 
El 

Ei 
B 

tli 

ES 




32 



SURVEYING 



standard, and one handle of a chain is held against it in testing. 
At every 10 ft. along the base a brass plate is sunk in flush with the 
granite with a score across, marking the distance. At the 100-ft. 
end a brass plate contains the score representing the true length 
of 100 ft., and is further divided into inches and decimals for a 
short distance on each side of the 100 mark to enable the error of 
the chain under test to be ascertained. 

When the instrument to be used is the chain, its length must be 
constantly checked. In ordinary use the joint rings and hooks 
of the links are liable to open and stretch, thus increasing its length, 
while it may be shortened by links getting bent, or the rings getting 
clogged with dirt or grass, and an error of 2 or 3 ins. in its length 

j2= 



ty- 




Fig. 35. — Field Standard for Testing Chain. 

may readily creep in. Where the chain is in constant use a pocket 
steel tape should be kept and compared with it every day. When 
the chain is too long a ring or two may be taken out. When too 
short, rings may be flattened, thus making them longer ; and when 
necessary, additional rings may be inserted. Where several rings 
require to be taken out or inserted, they should be distributed at 
intervals over the length of the chain, in order to avoid large error 
at any intermediate point. In all cases before testing a chain it 
should be carefully examined and cleaned, all bent links should be 
straightened, and kinks at the joints, if any, undone. 

Field Standard. — A convenient standard for testing a chain may 
be set up at the site of a survey by means of two stout pegs, A and B, 
driven into the ground at the proper distance apart and with nails 
fixed in their heads, as shown in Fig. 35. One handle of the chain 



CHAIN SURVEYING— FIELD OPERATIONS 33 

can be hooked on to the pair of nails on A, and when the chain 
is stretched out its other handle should be over the nail on 
peg B. With a standard arranged thus the chain can be tested 
and corrected by one person. 

Chaining on Level Ground. — The surveyor directs the operations 
and books the measurements. His two assistants handle the 
chain. The one at the forward end of the chain is called the 
" leader," the other is the " follower." It is assumed that the 
distance between the end stations of a survey line already ranged 
out is about to be taken, no intermediate measurements or offsets 
being required. The leader receives a full set of arrows and must 
count them to see that he has the correct number, viz., ten. If the 
leader is new to the work, the surveyor must instruct him as to 
the correct method of holding the chain handle, working the chain 
into line, and fixing the arrow. 

The link chain is generally kept gathered up into a sheaf and tied 
with a strap passing round its middle and through the two handles. 
To undo the chain, having taken off the strap, the chainman takes 
both handles in the left hand, allows some links next the handles 
to come loose, then takes the rest of the bundle in his right hand 
and throws it away from him. The chain will unfold itself and He 
along the ground double. If the two halves are entangled, an 
assistant should take hold of the 50 tally, pull the chain taut and 
untwist till the two halves are clear of each other. The chain may 
now be straightened out to its full length. Before using it, a careful 
inspection should be made to see that it is not kinked at any of the 
joints, and that none of the links are bent. The steel band is 
usually contained in a case and is undone by one assistant holding 
the case while the other pulls out the band. 

Before laying out the first chain length the leader should look 
along the line from the commencing station and note his direction, 
so that when he has drawn the chain out to full stretch he may 
not be far from the correct line. As the leader advances, the 
follower holds his end of the chain in his hand and imposes a gentle 
pull, carefully avoiding any jerk, when the leader has reached the 
limit of the first length. The leader stops, turns round and faces 
the follower, and, crouching down, holds the handle of the chain 
in his right hand, with an arrow held vertically against its end. 

S. D 



34 SURVEYING 

The point of the arrow should project an inch or two under the 
handle, and both should be held together out to the side so that 
no part of the leader's body may prevent the follower from seeing 
the arrow or the next forward pole. The follower meanwhile 
holds his end of the chain to the centre of the station mark, looks 
towards the forward pole, and by word or signal directs the leader 
to move the chain and arrow in one direction or the other, so as 
to bring them into line. The leader moves the chain to the side, 
and straightens it out by sending a wave along so as to lift it off 
the ground gradually from end to end, while simultaneously giving 
it a slight flick to the side and keeping a small amount of tension 
on it. It requires some experience to do this without transmitting 
an awkward jerk to the follower. 

When the 100-ft. chain is being used the following procedure may 
be preferable. The chain having been straightened out but not 
brought into correct line, the follower directs the leader to move 
his handle and arrow laterally until the arrow is in line. It will 
be short of its proper distance. The leader puts the point of 
the arrow into the ground, holds it there with his left hand and with 
his right hand straightens out the chain past the arrow and then 
moves the arrow forward to the end of the chain. On receiving 
the signal " Mark," or " Right," or " Down," from the follower, 
the leader presses the arrow firmly and vertically into the ground. 
The leader and follower then proceed, each holding an end of the 
chain. If there are two or more poles ranged out ahead, the 
leader should keep himself in line by their aid, and should count 
his paces so as to be ready to stop at once on feeling a gentle pull 
from the follower. When there is no pole left at the commencing 
station the leader should look back along the first chain after it 
has been lined in, and note some object in range therewith 
which may serve as a " back mark " in lining succeeding chain 
lengths. 

The follower has a record of the number of whole chain lengths 
measured in the number of arrows which he has picked up. When 
ten chain lengths have been measured and the leader has put in 
his last arrow he leaves the chain lying on the ground and waits 
until the follower comes up. The follower picks up the last arrow, 
taking care to mark the point by some other means, counts the 
arrows to see that he has ten, then hands them back to the leader. 



CHAIN SURVEYING— FIELD OPERATIONS 35 

The surveyor notes in his book the distance 1,000 (feet or links), 
the leader again pulls the chain ahead, and chaining proceeds as 
before, until the leader fixes an arrow within a chain length of the 
end station. Suppose that the total distance is 1,457 ft. The 
leader, having put in his fourth arrow, pulls the chain past the 
station until the follower reaches the arrow. The follower holds 
his handle to the arrow while the leader, having come back to the 
station, pulls the chain taut and reads the distance. The surveyor 
checks this, finds that the follower has four arrows, and makes 
sure that this tallies with the number retained by the leader, and 
sees from his book that ten complete chains have been already 
measured. The total distance is, therefore, 1,457 ft., which he 
enters in his book. 

The most common error in reading an intermediate distance 
on the chain arises from the fact that each division represents two 
different distances, according as the measurement is taken from 
one end or the other. Unless care is taken to read from the proper 
end an intermediate distance, such as 48 ft., may quite readily 
be read and booked as 52 ft., and while it is not so common to make 
the mistake of reading 71 when the distance should be 29, it is 
quite common to read 31 instead of 29. Mistakes are most readily 
made in reading points near the centre of the chain, and the resulting 
error, being comparatively small, is sometimes very difficult to 
locate. 

Precautions to be observed by the Leader and Follower. — To 

recapitulate shortly, the leader should observe the following 
precautions in chaining : — 

He should count his paces and be ready to stop when each 
chain length is out, and should stop at once on feeling a 
slight pull. 
He should guide himself almost into line. 
He should hold the arrow vertical and fix it vertical. 
He should carefully avoid tugging or pulling violently on the 
chain in getting it into fine. 
The following precautions should be observed by the follower : — 
He should stop the leader at the right moment by a gentle 

pull. 
He should never allow his handle to drag on the ground, 

d2 



36 SURVEYING 

nor pull up the leader with a jerk by putting his foot 
on it. 
In guiding the leader into line when there is only one mark 
ahead to line to he should be very careful to sight from a 
point directly above the arrow. This is important and 
difficult. 

Chaining on Sloping Ground. — If the line joining two points 
whose distance apart is to be determined is not level, the distance 
measured directly along the surface of the ground will not be the 
true horizontal distance between the points, but will be somewhat 
greater. Any distance so measured must, therefore, be corrected 
to allow for the slope, or some other method of measuring must be 



100' 




Fig. 36. — Measuring on Sloping Ground. 

used which will get rid of the effect of the slope. Three methods of 
measuring on sloping ground are illustrated in Fig. 36. 

Method of Stepping. — The first method is known as " stepping," 
and is best accomplished when chaining downhill. The follower 
holds his end of the chain firmly at the starting point A, while the 
leader stretches the chain, or a portion of it, out horizontally in the 
air in correct line as directed by the follower, and pulls it tight. 
The end of his handle comes to B, and using a plumb-bob he transfers 
this point vertically to the ground to point b and marks it with an 
arrow. The chain is then pulled forward, the follower holds his 
end to point b, and the same process is repeated. For accurate 
work the length of each step should not amount to 50 ft. when a 
heavy chain is used. Much better results can be got by lining out 
the chain full length down the slope, leaving it lying on the ground 
as a guide to the direction, and using a light steel tape for the 



CHAIN SUKVEYING— FIELD OPERATIONS 37 

stepping. This can be stretched nearly horizontal in a length of 
50 ft. The chief source of inaccuracy in stepping is due to the tape 
or chain not being held horizontal. Both the leader and the fol- 
lower are badly placed for estimating horizontal direction by the 
eye, and on steep ground even the surveyor standing apart may 
be deceived. The only reliable method is for the leader to raise 
or lower the tape until it makes a right angle with the plumb-line 
as nearly as can be judged. 

In stepping down a slope in short lengths it is desirable that the 
leader should mark the intermediate points between the ends of 
each chain by means of twigs or in some other temporary manner. 
If he uses arrows for the intermediate points, he must be careful 
to get them back from the follower at the end of each whole chain 
length, otherwise a miscount of the total number of chain lengths 
will almost surely result. 

Method by Measuring on Slope and applying Correction to Total 
Length. — The second method consists in measuring the total length 
between the points along the sloping surface of the ground, and 
thereafter calculating the deduction which must be made to reduce 
this length to the true horizontal distance. In Fig. 36, AC repre- 
sents a chain length measured down the slope, and Ac is the 
corresponding true horizontal distance, while cB represents the 
amount to be deducted from AC to arrive at the true horizontal 
distance. 

Let a = angle of slope. 

Then Ac = AC cos a. 

and cB = AC — Ac = AC (1 — cos a). 

If, therefore, a total length L has been measured along a slope 
of angle a, the true horizontal length will be L cos a or the 
length measured will be too great by the amount L (1 — cos a). 
If the length between two points consists of several stretches with 
different slopes, a separate calculation must be made for each 
stretch. This method of measuring on sloping ground is only 
convenient where there are no intermediate points whose distances 
require to be recorded. 

The table on p. 38 gives the true horizontal length corresponding 
to unit distance measured on slopes up to 30°. 



38 SURVEYING 

Table of Horizontal Lengths corresponding to Unit 
Lengths measured on Slope. 



Slope in 


Horizontal 


Slope in 


Horizontal 


Slope in 


Horizontal 


Degrees. 


Length. 


Degrees. 


Length. 


Degrees. 


Length. 


1 


•99985 


11 


•98163 


21 


•93358 


2 


•99939 


12 


•97815 


22 


•92718 


3 


•99863 


13 


•97437 


23 


•92050 


4 


•99756 


14 


•97030 


24 


•91355 


5 


•99619 


15 


•96593 


25 


•90631 


6 


•99452 


16 


•96126 


26 


•89879 


7 


•99255 


17 


•95630 


27 


•89101 


8 


•99027 


18 


•95106 


28 


•88295 


9 


•98769 


19 


•94552 


29 


•87462 


10 


•98481 


20 


•93969 


30 


•86603 



Example. — In measuring a total distance of 667 ft. along the 
ground between two points, the first 200 ft. were on a slope of 5°, 
the next 300 ft. on a slope of 8°, and the remaining 167 ft. on a 
slope of 6°. Using the above table, find the true horizontal dis- 
tance between the points. 

For 5° slope, -99619 X 200 = 199-238 
„ 8° „ -99027 X 300 = 297-081 
„ 6° „ -99452 X 167 = 166-085 



Horizontal distance = 662-404 



Method by Measuring on Slope and applying Correction on Ground 
at each Chain. — In this method the chain is laid out down the slope, 
and an arrow is put in temporarily at C (Fig. 36). The slope angle 
having been measured with the inclinometer, the additional dis- 
tance C6 required to make A6 have a true horizontal length of 
100 ft. or links can be calculated or found from a table 

A6 = AB sec a = AC sec a 
.-. C6 = AC sec a — AC 
= AC (sec a — 1). 

The distance C6 having been found, it is measured off on the 
ground by means of tape or foot-rule, and the arrow is advanced 



CHAIN SURVEYING— FIELD OPERATIONS 



39 



to b. Each, chain length is similarly corrected, care being taken to 
determine the slope anew at each change of inclination. This 
method is generally more convenient than stepping where the 
slopes are moderate and the ground fairly uniform. The slope 
angle is determined closely enough by an inclinometer of the hinged 
type placed on a pole or offset staff laid on the ground. 

The following table shows the corrections required per chain 
length of 100 ft. or 66 ft. for slopes up to 20° :— 

Table op Corrections to be made on the Ground for 
Chain Lengths measured on Slope. 





Correction to be made on ground, measured on the slope. 


Slope Angle. 








Degrees. 


Feet per 100 ft. or 
links per 100 links. 


Per 100 ft. 


Per 66 ft. 






Ft. Ins. 


Ft. Ins. 


1 . 


•01523 


o oa 


0| 


2 . 




•06095 


Of 


OJ 


3 . 




•13723 


If 


11 


4 . 




•24419 


21| 


o Hf 


5 . 




•38198 


4 T 9 ¥ 


3 


6 . 




•55083 


6f 


4f 


7 . 




•75098 


9 


5|f 


8 . 




•98276 


11*£ 


7| 


9 . 




1-24651 


1 2H 


9| 


10 . 




1-54266 


1 6| 


1 0£ 


11 . 




1-87167 


1 10^ 


1 2-if 


12 . 




2-23406 


2 2if 


1 5H 


13 . 




2-63041 


2 7A 


i m 


14 . 




3-06136 


3 Of 


2 0i 


15 . 




3-52762 


3 6| 


2 3ff 


16 . 




4-02994 


4 Of 


2 m 


17 . 




4-56918 


4 6H 


3 T \ 


18 . 




5-14622 


5 If 


3 4| 


19 . 




5-76207 


5 n 


3 9f 


20 . 




6-41778 


6 5 


4 2|f 



CHAPTER IV 

CHAIN SURVEYING — RUNNING A SURVEY LINE 

In this chapter consideration is given to the methods of fixing 
points in relation to a survey line by means of offsets and ties. 
The measurements required for properly fixing buildings, irregular 
boundaries, &c, are then dealt with, and the running of a survey 
line is described in detail. Methods of booking the work are 
described and examples are given, illustrating the booking of 
survey lines. 

Fixing Positions of Objects relative to Points on a Survey Line. — 

The determination of the positions of objects in relation to points 
fixed by measurement along a survey fine usually constitutes 
the bulk of the detail work in surveying. The methods adopted 
for a survey line in a chain survey apply equally to a survey line 
in a triangulation or traverse survey. Two methods are in common 
use. In the first method, that of " offsets," the point is located 
by the measurement of a distance and an angle (usually 90°) from 
a point on the chain line. 

In the second method, known as the method of " ties," the 
distance to the object is measured from two separate points on the 
chain fine, the three points forming as nearly as possible an equi- 
lateral triangle. 

The two methods applied to fixing the corners of a building 
are illustrated in Figs. 37 and 38. In Fig. 37 the corners A and B 
of a building are fixed by perpendicular offsets of 30 ft. ins. and 
20 ft. ins. from points at distances 530f and 564| respectively 
along the chain fine. If the building were rectangular, the only 
further measurement required to enable its position to be plotted 
on paper would be its width. The surveyor would, however, have 
no check on the accuracy of the work plotted from these measure- 
ments, and a mistake made either in the field measurements or in 
the plotting would pass unnoticed. By measuring and recording 



CHAIN SURVEYING— RUNNING A SURVEY LINE 41 

the length AB a check on the accuracy of the plotted points A 
and B would be obtained, and a check on the directions of the sides 
AD and BC would be got by noting the points where these directions 
cut the chain line. In general, to obtain satisfactory results, in 
work which has to be plotted to a fairly large scale, all the measure- 
ments indicated in Fig. 37 should be recorded. 

In Fig. 38 the method of fixing the building by means of ties is 
illustrated. Point E is located by the lengths 31 ft. 6 ins. and 
33 ft. 3 ins., measured from points at distances 522 and 545 respec- 





Fig. 37.— Offsets. 



Fig. 38.— Ties. 



tively along the chain line, and point F is similarly fixed by two 
measurements. To avoid recording too many points along the 
chain line, point 522 is chosen in line with the end of the house, 
and similarly point 545 is made to do double duty. The complete 
measurements which should be recorded are as shown in the 
figure. 

A comparison of the two figures shows that the method by offsets 
involves less measuring on the ground and less sketching and figuring 
in the note book. The comparative accuracy of the methods will 
depend almost entirely on the accuracy with which the right angles 
in the offset method are measured. With a reliable optical square 



42 



SURVEYING 



or prism properly used the offset method will be as accurate as the 
tie method, while, given equally accurate work in both cases, the 
plotting of offsets will involve less error than the plotting of ties 
and will at the same time be more expeditiously carried out. 




Survey 

Fig. 39. — Finding Correct Offset Distance. 

The extent to which the perpendicularity of right angle offsets may 
be estimated by the eye will depend in part on the scale to which the 
plan is to be plotted. On a scale of 20 ft. to the inch distances smaller 
than 3 ins. are scarcely appreciable on the paper, and for scales 
between 20 ft. and 80 ft. to an inch, it is customary to record chain 
distances, offsets and ties to the 
nearest quarter foot or half link. A 
person with an average eye will 
easily notice that one line is not 
perpendicular to another if the 
deviation amounts to 1 ft. in forty. 
This would indicate that the length 
of offset estimated by eye should not 
exceed 10 ft. if the error in location 
of the point is to be under 3 ins. 
For small scales, such as the ^Joq, 
where distances under 2 ft. are 
scarcely appreciable on the paper, 
longer offsets may be set out by the eye. The method of pro- 
cedure in finding the length and position of a perpendicular offset 
is as indicated in Fig. 39. The length of the offset is the shortest 
distance from object to chain obtained by swinging the tape about 
the object as centre. The position of the offset on the chain will, 
in general, be as accurately determined by noting the point where 
the tape makes symmetrical angles with the chain. 




Fig. 40. — Accurate Short 
Offsets. 



CHAIN SURVEYING— RUNNING A SURVEY LINE 43 




\ 



/ 
/ 
/ 

/ 
/ 
I 
i 

\ 
\ 



.5 ■+= 
-3 js 



The method of setting out and measuring accurate short offsets 
by means of a large wooden set-square 
and an offset staff, as sometimes practised 
in town surveying, is illustrated in Fig. 40. 



Points to which Offsets or Ties should 
be Taken. — The position of any straight 
line relative to a survey line is absolutely 
determined if its two end points are 
correctly fixed by offsets or ties. Any 
line or boundary, therefore, which con- 
sists of a series of straights will be com- 
pletely located by taking offsets or ties 
to all the angles. In Fig. 41 the straight 
portions of boundary AC and CD are 
completely determined by the offsets 
taken to the points A, C, and D. Any 
straight line, whose ends have been 
fixed, may be used as a subsidiary base 
line to which intermediate points and 
adjacent objects may be referred. For 
example, it may be more convenient to 
determine the point B on the straight 
line AC by a measurement of the distance 
AB than by an offset or by ties from 
the survey line. 

Curving lines or boundaries are sur- 
veyed by taking offsets or ties to points 
chosen at intervals along the curve, the 
spacing being such that a fair curve 
drawn through the points plotted on 
paper will not differ appreciably from 
the true form of the boundary. The 
flatter and more regular the curve the 
greater may be the interval between off- 
sets, while the larger the scale of the plan \ 
the smaller should be the interval. A 
railway curve of uniform curvature may be accurately 
in, with the use of manufactured curves as rulers, from 



drawn 
points 



44 



SURVEYING 



surveyed at fairly wide intervals, provided particular care is 
taken to fix the end portions sufficiently minutely, as the curva- 
ture is often variable near the tangent points. In surveying 
regularly curving objects it is most convenient, both as regards 
booking and plotting, to take the offsets at equal intervals, say 
every chain length or every 20 or 50 ft. or links, or other round 
distance, as the case may require. In Fig. 41 positions of offsets 
are indicated for a curving boundary, DEF. Junction points with 
other boundaries, such as point E, must be carefully and accurately 
located, as well as all definite angles, such as point F. 




178 




Fig. 42. — Fixing a Building. 



Fig. 43. — Complete Measure- 
ments to fix a Building. 



Fixing Buildings, &c— Detached buildings and houses commonly 
consist of rectangular forms in plan. In surveying such buildings, 
where they are of simple outline, it will usually be preferable to 
carefully fix one of the longer sides of the building in relation to the 
survey line and to locate the rest of the building in such a way 
that it can be plotted from this side as a base. Sufficient measure- 
ments must be taken to check the directions which the sides make 
with each other. Fig. 42 shows a rectangular building lying with a 
long side AB adjacent to the survey line. Points A and B would 
be fixed by perpendicular offsets or by ties. The further measure- 
ments necessary to enable the building to be plotted are the width 






CHAIN SURVEYING— RUNNING A SURVEY LINE 45 




Fig. 44. 



-Fixing Building from 
Short Side. 



of the main building and the length and width of the projection at 
the back. The plotted points A and B fix the long side of the 
building on the paper and furnish a base line from which the remain- 
ing sides of the building may- 
be plotted. 

The measurements shown 
in Fig. 42, if correctly made, 
would enable a rectangular 
building to be accurately 
plotted. A mistake in any of 
the measurements would, how- 
ever, remain undetected. To 
check the accuracy of the 
work and to enable the surveyor to have confidence in his results, 
all the measurements shown in Fig. 43 should be taken and 
recorded. The measured length of AB then furnishes a check on 
the accuracy of the plotted points, and the true directions of the 
ends of the building are established by noting the points where 

these directions cut the sur- 
vey line. The accuracy of 
the various sides is to a large 
extent proved by taking the 
separate measurement of each. 
It may lead to error to assume 
that corresponding sides are 
equal. 

If the side adjacent to the 
survey line is short in com- 
parison with the side of the 
building it will not be suffi- 
ciently accurate simply to fix 
that side in relation to the 
survey line. The dotted fines 
in Fig. 44 indicate the measurements which are sufficient, 
theoretically, to completely fix the side CD and the directions of 
the two adjacent sides. The location of the building from these 
measurements would not, however, be reliable, as the length of CD 
is too short in comparison with the size of the building to furnish 
a proper base. A small error at C or D would cause a magnified 




Fb 



Fig. 45. 



-Fixing Building from 
Long Side. 



46 



SURVEYING 



displacement of the other end of the building. It is, therefore, 
advisable in such a case to proceed as shown in Fig. 45, where the 
position and direction of the long side DE are fixed by the ties 
FE and GE. The rest of the building may then be plotted from DE 
as a base, with small possibility of error. 

In the case of irregular-shaped buildings it will seldom happen 
that all the essential corners can be fixed directly from the survey 
line. The shape of the building will in many cases not be definitely 
fixed, even when all the corners adjacent to the survey line have 
been located by offsets or ties and the measurements of all the sides 





B -® 

Figs. 46 and 47. — Fixing Irregular Buildings. 

have been taken. Corners which are unattainable from a main 
survey line will often be best fixed from a subsidiary survey line 
specially set out for the purpose. Under suitable circumstances 
subsidiary fines may be set out to encompass the building. Fig. 46 
shows an irregular-shaped building and indicates a method by which 
it might be completely surveyed. The front corners and the ends 
of the building are fixed by the offsets and ties from the survey line. 
The tie lines used in locating the ends of the building serve to fix 
the points A and B. A line run between these points serves as a 
subsidiary survey line, from which to fix the remaining corners 
by offsets or ties. The measured lengths of the sides of the building 
form a check on the accuracy of the plotted points. 



CHAIN SURVEYING— RUNNING A SURVEY LINE 47 

Fig. 47 shows how the optical square may be utilised in arranging 
a system of subsidiary lines to enclose an irregular building. Lines 
AB and CD are set off at right angles to the survey lines and serve 
along with the line run from B to D as base lines, from which all 
the essential corners can be located. In the figure the various 
corners are shown fixed by means of perpendicular offsets. In 
practice, the methods of fixing the points will depend largely on 
the circumstances of each case. A point which, owing to some 
obstruction of view or otherwise, cannot be determined by a per- 
pendicular offset may often be readily fixed by ties and vice versa. 

Locating Irregular Boundaries, &c. — A portion of an irregular 
boundary may consist of short lengths of straight. In that case, 
as already indicated, offsets would be taken to each corner formed 
by the junction of two straights. In surveying boundaries which 



Fig. 48. — Offsets to Irregular Boundary. 

are completely irregular, as indicated in Fig. 48, the same principle 
is applied. Points of abrupt change are chosen, dividing up the 
boundary into portions which, apart from minor irregularities, 
may be considered as sensibly straight. Offsets are taken to these 
points, and the forms of the intervening sinuosities are sketched in 
the note book as accurately as possible. 

The amount of care which should be expended in the location 
will depend on the nature of the boundary. In surveying definite 
and permanent property divisions, however irregular, the offsets 
and measurements should be so taken that there will be no room 
for appreciable variation between the plotted form and the actual. 
In the case of variable outlines, such as the margin of rivers and 
lakes, whose position depends on the level of the water, and like- 
wise in the case of indefinite boundaries, such as the edge of marshes, 
&c, while the broad features of the boundary should be accurately 
determined, there is no need for great refinement in fixing minor 
details. 



48 SURVEYING 

Length of Offsets. — The closer the chain line lies to the objects 
and features which are to be surveyed, the more accurately and 
expeditiously will the work be accomplished. The ordinary linen 
tape with which offsets and detail measurements are made is not a 
very reliable instrument. An error of 3 ins. is not uncommon in a 
tape 66 ft. long, and hence one reason for limiting the length of 
offsets to as small a size as possible. Measurements which exceed 
one tape length cause extra trouble and delay, so that the survey 
line should preferably never be so far from the work as to cause 
the length of a tie line or offset to exceed the length of the tape. 
The aim should always be to make the offsets as small as possible, 
and it will often be true economy to lay out some additional 
survey lines to effect this. A case in point often occurs in 
surveying a wide road with detached houses on each side. A 
single survey line might, with the use of long offsets and ties, 
suffice to pick up both sides of the road and the fronts of the 
houses. It would be generally preferable, however, even apart 
from considerations of interference due to traffic, to lay out 
a survey line on each side of the road and to survey each side 
separately. The booking of the notes is thereby simplified, 
and likewise the plotting, and there is less chance of omission 
and error. 

Running a Survey Line. — The methods of ranging out a line 
and measuring its length have been already explained. In running 
a survey line, that is, in performing simultaneously the operations 
of chaining the line and locating objects from it, the chain is left 
lying lined out straight on the ground at each length until all the 
necessary offsets, ties, and other measurements have been taken. 
Where several offsets or ties require to be measured from the same 
chain length, they should be taken systematically in order, working 
from the rear towards the forward end of the chain. Mistakes 
in reading the chain and in booking are apt to occur if the offsets 
are taken out of consecutive order or working backwards. AVhere 
numerous offsets occur on each side of the chain, it is sometimes 
advisable to take first all the offsets and measurements on one side 
of the chain and then all those on the other side. Figs. 51 to 
53 show some examples taken from actual practice of lines run 
in surveying various commonly occurring classes of features. 



CHAIN SURVEYING— RUNNING A SURVEY LINE 49 

They give an idea of the offsets and measurements required and the 
methods of booking. 

Field Book. — The main requirements of the field note book 
are that it should contain good quality stout opaque paper, should 
be well bound and of a size convenient for the pocket. Common 
sizes for field books run from 7 ins. by 4§ ins. to 9 ins. by 5| ins. 
The paper may be either plain or ruled with a faint red line down 
the centre of each page to represent the survey fine. Instead of 
a single line two parallel fines a short distance apart are sometimes 
used, the column thus formed being reserved for the insertion 
of all distances along the survey line. Any chance of confusion 
between these distances and the offset or other measurements is 
thereby avoided. The plain paper field book can be utilised for 




Fig. 49. 



Fig. 50. 



all kinds of note-taking, and is more convenient than the ruled 
form if a large amount of general sketching is required. When a 
survey line is to be booked a pencil fine is ruled on the paper to 
represent the survey line. If a considerable width of ground is 
being surveyed all on one side of the chain, it may be advisable to 
rule the pencil line near one edge of the page so as to get sufficient 
room for the notes. Where the bulk of the work consists of 
recording the details of survey lines the note book with single 
ruled line is recommended. The book may open either the short 
way or the long way, as indicated in Figs. 49 and 50. In the 
latter case the line is continuous from side to side and the two 
pages become practically one, and are usually so numbered. 

On the inner front page the surveyor's name and address should 
be written. The date and the names or initials of the members 
of the party should be inserted at the top of each page on which a 
fresh day's work is commenced. A few pages at the beginning 



50 SURVEYING 

of the book should be reserved for reference diagrams of the survey 
lines and an index giving the page or pages on which each line is 
detailed. 

Field Notes. — The importance of making the field notes clear, 
accurate, and complete cannot be too much emphasised. A good 
quality pencil should be used, kept well pointed. It should not 
be so soft as to smear the paper when the hand is brushed across 
the page, nor so hard as to make but faint marks. A pencil of 
degree H or F will usually be found suitable. Figures should be 
plain and well formed. Whenever dubiety would be likely to arise 
as to the points between which a figured distance extends, the 
proper points should be clearly indicated by means of arrow heads 
or otherwise. The clearness and neatness of the work is greatly 
enhanced if all notes, names and explanations, &c, are neatly 
printed in italics or block. With practice neat sloping italic 
printing can be done almost as quickly as writing. The sketches 
should be arranged of a size such that there will be ample room 
for all figures. Crowded figures are a fruitful source of confusion 
and mistakes in plotting. 

Without attempting to draw the objects surveyed to scale the 
endeavour should be made to have them set down in fairly recog- 
nisable proportion and in good relationship to the survey fine. 
Straight fines should be shown straight, all definite angles and 
corners should be clearly indicated as such, points where a straight 
outline changes to a curving one should be carefully marked, and 
the sketch of an irregular boundary should be as true to actual 
form as possible. It will be found advantageous in sketching to 
make the following exaggerations as compared with a true to scale 
plan : Magnify the size at points of minute detail, allowing in every 
case sufficient room for the dimensions. Decrease the length of 
all long straight fines. Where two straight lines meet at an angle 
of nearly 180°, so as to be nearly in one line, draw them at a more 
acute angle in the sketch and make the position of the corner quite 
definite. A fence or boundary line, &c, which crosses the survey 
line nearly, but not quite at right angles, should be drawn on the 
sketch with an exaggerated inclination to the perpendicular in the 
proper direction. When, as sometimes happens, it is unavoidable 
that an actual straight line should be represented on the sketch 



CHAIN SURVEYING— RUNNING A SURVEY LINE 51 



****<&>*'> 




{id* ^ 



M3/S A 



Fig. 51. — Example of Survey Line. 



e 2 



52 SURVEYING 

by a crooked line, write the word " straight " alongside and indicate 
the extent by arrow heads. This is most often necessary where 
straight lines cross the survey line. 

Examples of Survey Lines. — Fig. 51 shows a page extracted from 
a survey book to illustrate the method of booking perpendicular 
offsets. It is typical of much of the kind of surveying required in 
open cultivated country. The straight line running up the page 
represents the survey line, the commencement being at the bottom 
of the page. The objects to be surveyed are sketched about this 
line in the position which they occupy relative to the survey line 
and the distances along the survey line, and the lengths of the 
perpendicular offsets to the defining points are figured on. Where 
offsets to several points are taken from the same point on the chain 
line, the distance to each point should be measured from the chain. 
Thus, in the figure, where three offsets are shown from the same 
point on the chain, the lengths figured are the total measurements 
from the chainline to each point, not the separate distances from 
point to point. 

Fig. 52 shows the notes of a survey line on a curving railway 
embankment. The stations are fixed clear of the running lines, 
so that poles may be planted for lining out purposes without risk 
of disturbance from trains, and in order that the theodolite, if used, 
may be safely set up. In railway work the boundaries, railway 
lines, and main structures are of primary importance, while edges 
of slopes and such like are of less importance and need not be sur- 
veyed in great refinement. Since the rails in a double track are 
parallel it is only necessary to survey one of the rails and to measure 
once and for all the widths of the spaces between the rails. To 
facilitate plotting, the offsets should be taken to the same rail 
throughout the length of each survey line. When the surveyed 
rail has been plotted on the plan the others which are parallel to it 
may be drawn in at the correct distance apart. 

Fig. 53 shows the notes of a survey line locating an irregular 
stream and boundary fence. The offsets to fix the stream are 
taken at the points which best determine the changes of direction 
of its banks. Minor irregularities between offset points should be 
recorded by careful sketching only. 






CHAIN SURVEYING— RUNNING A SURVEY LINE 53 




V 

Fig. 52. — Example of Survey Line — Bailway Lines, 



51 



SUEVEY1NG 




Fig, 53. — Example of Survey Line — Stream, etc, 



CHAPTER V 

CHAIN SURVEYING — ARRANGEMENT OF SURVEY LINES 

The principles governing the laying out of a system of lines 
suitable for surveying a small area are dealt with, in this chapter, 
and examples of desirable arrangements of lines are given. No 
hard and fast rules as to arrangement of survey lines can be given, 
as this is largely controlled by the configuration and features of 
each area. 

Arrangement of Survey Lines. — The framework of survey lines 
forming the basis of a chain survey must be laid out as a con- 





FlG. 54. — Displacement of a 
Point due to Error in Side. 



Fig. 



55. — Limits of Well-conditioned 
Triangle. 



nected system of triangles. To get good results in plotting, the 
triangles should be as nearly equilateral as possible. To exclude 
the possibility of large error due to ill-conditioned triangles, it 
should be made a rule that no angle of a triangle should be less 
than 30°. 

Let A and B (Fig. 54) represent two points plotted in correct 
relation to each other, and let C represent the correct position 
of the third point, whose position has been fixed by measuring the 



56 SURVEYING 

distances AC and BC. If, due to a mistake or error, the length AD 
has been used instead of the true length AC, the plotted point will 
be at C at a distance C'C from its true position. An inspection of 
the figure shows that for an error confined to one of the sides the 
resulting displacement of the plotted point can never be less than 
the error. Practically, if the angle at C is 90° the displacement of 
C will be just equal to the error in the side AC ; for an angle of 
60° the displacement will be 1-15 times the error, and for an angle 
of 30° the displacement will be equal to twice the error. For 
angles smaller than 30° the displacement will be still greater. 
If the chance of error is confined to two sides of the triangle the 
best result will be obtained when these sides make an angle of 90° 
with each other. If, however, there is equal liability to error in all 




Fig. 56. —Proof Line for Triangle. 

three sides of a triangle, the best form is the equilateral. The 
condition that no angle of a triangle should be less than 30° (which 
necessarily involves that no angle should be greater than 120°) 
means that the third angle of a triangle on the base AB (Fig. 55), 
may be anywhere within the hatched area. 

The endeavour should be to have as few main triangles 
in the framework as possible and to have them well checked 
by proof lines. A proof line is a line which serves to test 
or rather confirm the accuracy of a triangle. Take the case of a 
single triangular field surveyed by a single triangle, as shown in 
Fig. 56. A mistake in the measurement of any of the sides would 
cause the field to be wrongly plotted, and the mistake would not be 
detected unless a check measurement be made. The measurement 
of the length of the line from C to a point D fixed by measurement 
along the line AB would form such a check in the case of the given 



CHAIN SURVEYING— ARRANGEMENT OF LINES 57 



triangle. If after plotting the triangle ABC the scaled length 
of CD is found to agree with the length measured on the ground, 
then it may reasonably be assumed that the surveying and plotting 
are correct. It should be clearly noted that a proof line, such as 
CD, does not form an absolute test of the accuracy of the work, 
as any errors in the measurement of the sides which caused a dis- 
placement, however considerable, of point C along the arc EF 
struck from centre D, would remain undetected. 

The main survey lines should be arranged to pick up as much 
of the detail as possible, and should, as far as practicable, approach 
most closely to the more important portions of the work. The 
condition as to well-shaped triangles may require some of the main 
lines to be set out without regard to picking up detail, and such 




Fig. 57. — Single Triangle forming Basis of Survey Lines. 

lines should be arranged with due consideration to the necessity 
for connecting subsidiary lines to them. An existing plan of an 
area to be surveyed is of considerable service in the preliminary 
work. The proposed main stations being marked on roughly in 
position, it can be seen at a glance whether the system of triangles 
is good or indifferent. It is an instructive exercise for the beginner 
in surveying to take a good existing map of a small area, and, 
without going to the ground, lay down thereon a system of lines 
which in his judgment would be suitable for surveying the features 
shown. Let him then proceed to the area and examine whether 
his lines are practicable. It will be surprising if his arrangement 
at all suits the actual conditions, and the comparison should impress 
on him the extent to which the layout of survey lines is governed 
by the configuration of the ground and minor natural features 
which affect the lines of view. 



58 



SURVEYING 



Fig. 57 indicates how a single main triangle may serve as the 
basis for surveying an area of several enclosures. The subsidiary 
lines DE and GP check the accuracy of the main triangle, and at 
the same time serve to locate the interior boundaries. Line FG 




Fig. 58. — Survey Lines for Quadrilateral Enclosure. 

is produced to H, and the accuracy of this point is checked 
byBH. 

To survey a quadrilateral enclosure two triangles will usually 
be required. In the case illustrated in Fig. 58 the triangles which 
would be plotted are ABC and ADC. Line BD forms the check 




Pig. 59. — Survey Lines for Quadrilateral Enclosure. 

line. Instead of proceeding as above, point D might be deter 
mined by laying down the triangle BDC on the base line BC, but as 
BDC is a badly conditioned triangle the accuracy of the plotting 
would not be reliable. Similar considerations would indicate in the 
case of Fig. 59 that the triangles ABC and BDC should be plotted, 
the diagonal AD serving as check line. 









CHAIN SURVEYING— ARRANGEMENT OF LINES 59 

For more extensive surveys an arrangement of triangles laid out 
on each side of a straight line running through the area is to be 
desired. Such an arrangement is shown in Fig. 60. Line AB 
represents the through base line. This would be laid down first 
on the paper. The triangles shown in full lines would then be plotted. 
Check lines and subsidiary survey lines are shown dotted. 

Marking Survey Stations. — The survey stations require to be 
marked in a more or less permanent manner, depending on the 
extent and duration of the survey. In soft ground wooden pegs, 
and in hard roads or streets pointed iron spikes or nails are most 




-Arrangement of Triangles. 

convenient for the purpose. They should be driven till flush with 
the surface wherever there is traffic. Wooden pegs may be rendered 
more conspicuous and easily found by cutting away the turf around 
their heads. 

Referencing Survey Stations. — The principal stations should be 
carefully " referenced," so that their positions may be recovered 
should the pegs themselves happen to get removed. A station 
may be referenced by taking accurate measurements to it from two 
definite points on adjacent permanent objects. The more nearly 
the fines joining the peg to the reference points intersect at right 
angles, the more definitely will the station be fixed. Stations at 
a distance from any permanent objects may be referenced by driving 



60 SURVEYING 

two additional pegs, one on each side of the station and some distance 
away from it, and so that all three pegs are in line. The distance 
from each side peg to the station is noted. To avoid confusion 
the reference pegs should be distinct in character from the station 
Peg- 

Numbering Stations. — A diagram of the survey lines with main 
stations numbered should be inserted in the beginning of the field 
note book. Minor stations along the main lines, which are usually 
fixed as the work proceeds, may be given the number of the nearest 
main station with a distinguishing letter, as 9a, 13c, &c. Each 
survey line, as booked in detail in the field book, should have the 
numbers of the commencing, terminal and intermediate stations 
clearly marked to correspond with the numbers on the reference 
diagram. A line commencing, say, at station 6 and terminating 
at station 7 is designated Line 6 — 7, and this should be plainly 
written or printed at the top of the page containing the notes of 
this line, and similarly for all the lines. It is a convenience to have 
the lengths of all main lines marked on the reference diagram. 






CHAPTER VI 

CHAIN SURVEYING — ERRORS 

Only from a careful study of the errors which may arise in the 
various operations of surveying and the relative magnitudes of 
these errors can a surveyor attain to a proper appreciation of the 
relative importance of the operations and be enabled to arrange 
his work to the best advantage, having due regard to accuracy 
and economy. This chapter deals with the principal errors which 
may arise in the operations of chain surveying, including errors in 
measuring the lengths of lines resulting from various causes, errors 
in ranging out the survey lines, and errors in locating objects 
and details. 

Errors in Measuring a Length. — The important sources of error in 
linear measurements are : — 

(a) Incorrect length of chain and incorrect graduation. 

(6) Chain not held horizontal, or wrong allowance made for 
slope. 

(c) Chain not stretched straight and tight between its ends. 

(d) Sag on chain. 

(e) Arrows wrongly fixed at end of chain. Chain not correctly 
held against arrow. 

( / ) Mistakes in reading the chain and in recording distances. 

Error in Length of Chain. — A new steel band should be tested on 
an official standard. At a temperature of 60° and under a pull of 
15 lbs. the error of a steel band should not exceed \ in. Excluding 
the slight alterations due to variation of temperature and pull, 
a band may be considered as of constant length for all purposes 
for which a chain survey is likely to be required. A very slight 
lengthening may occur, due to wear at the joints of the handles, or 
from splicing a break. 

The steel chain, as already indicated, is of very variable length. 



62 SUKVEYING 

A given pull causes a much larger stretch than in the case of the 
band, and alterations due to variation of pull may be a considerable 
source of error. The chain when in use continually changes in 
length, and these alterations are an important source of error. 

It is important to distinguish between the effect of a constant 
error and that of a variable error on the accuracy of a survey. 
Suppose that a survey has been made with a band 101 ft. long 
divided into 100 equal parts supposed to be feet. The error in the 
length of the band will not affect the plotting or consistent accuracy 
of the work. If the scale of the plan be drawn on the assumption 
that the band is 100 ft. long, all measurements made by this scale 
will be wrong to the extent of 1 ft. in every 100 ft. If, on discovery 
of the error in the band, the scale be altered so that the distance 
which formerly represented 100 ft. be now made to represent 101 ft. 
the plan will be as accurate as if it had been made with a correct 
band. 

The whole effect of a constant error in the measuring instrument 
is to cause the scale of the plan to be slightly wrong. If the extent 
of the constant error is known, the scale can be altered propor- 
tionately so as to make the plan correct. 

The case is different if one portion of a survey be made with an 
incorrect chain and another portion with a correct chain. It will 
not now be possible to plot a correct plan, as the two portions are 
inconsistent. The same result is produced if the measuring instru- 
ment, such as the steel link chain, is subject to variable error, 
that is if its length alters during the course of the work. In im- 
portant work, if the error is to be kept to the lowest possible limit, 
the chain should be made correct before starting each day's work, 
checked on a reliable standard at the end of the day, its error, 
if any, noted, and when necessary a variable correction applied 
to the lengths chained during the day, proceeding on the assumption 
that the error has increased gradually. 

It should be carefully noted in making the corrections that if 
the chain is too long, all lengths measured will be too short, while 
if the chain is too short, the measured length of any distance will be 
too great. Inattention to this may result in the error of a length 
being doubled, in the belief that it is being eliminated. 

Kule to find the correct length of a line measured with an incorrect 
chain or tape : — 






CHAIN SURVEYING— ERRORS 63 

Multiply the measured length by the ratio of the incorrect length 
of the chain to the correct length. 

Suppose that the distance as measured between two points with 

a chain which was f in. too long, was 788-3 ft. To find the true 

distance — 

„ . Incorrect length 100*06 

Ratio - -: -f- - — — - 1-0006. 

Correct length 100 

True distance = 788-3 X 1-0006 = 788-8 ft. 

The above result will be correct if the chain is graduated in even 
divisions. If the chain is irregularly graduated the total error will 
consist of two portions — one due to the error in length of the chain 
and proportional to the number of whole chain lengths measured, 
the other due to the incorrect graduation and occurring in the final 
fraction of a chain length. Graduation errors might be positive at 
one part of the chain and negative at the other. 

Error due to Chain or Tape not being stretched Horizontal. — Any 

measurement taken on the slope is greater than the true horizontal 
measurement. The result, therefore, of not holding the chain 
truly horizontal, or of neglecting to allow for the effect of slope, 
is to cause the measurement to be too big. The error in a distance 
measured along a uniform slope will be proportional to the distance. 
As no surface other than a level one can have the same slope 
in all directions, the general result of neglecting slope in chain 
surveying will be to introduce variable error (always positive) in 
the lengths of the lines, with adverse effect on the accuracy of the 
work. 

The tables in Chapter III. show the extent of the error intro- 
duced in measuring on various slopes. The limit up to which a 
slope may in practice be treated as level can be determined from 
an examination of the table, keeping in view the purpose of the 
survey and the accuracy required. 

Errors from Chain not being stretched Straight and from Sag on 
Chain. — These two sources of error give rise to effects similar to 
those produced by neglecting slope. The error is always positive, 
that is, the measured distances are too great. With a 100-ft. chain 
a sag of about 8 ins. below the line joining the ends, or a like hori- 
zontal deviation at its middle when lying on the ground, will cause 



64 SURVEYING 

an error of \ in. When ordinary care is exercised in straightening 
the chain there is no room for appreciable error due to this cause. 
The case where the chain requires to be stretched across a wide 
opening with considerable sag seldom occurs. As the length, 
especially with a link chain, is considerably affected by the increased 
pull required, the most satisfactory method of getting the true length 
is to note the amount of sag and afterwards hang the chain with the 
same amount of sag in a position where the horizontal distance 
between its ends can be accurately measured. This can be readily 
effected on the side of a wall. 

Errors in Fixing Arrows and Marking Chain Lengths. — On level 
ground, where the handles can be held down on the surface, there 
should be little error in fixing the arrows or otherwise marking the 
chain lengths. When, however, owing to irregular ground, rank 
growth, slope, &c, the handle cannot be held to the ground the 



r- 85 ^ 




Pig. 61. — Error from Careless Fixing of Arrow. 

chance of error is greatly increased. If the leader, holding the 
top of his arrow against the handle, sticks it into the ground some- 
what off the perpendicular, and the follower thereafter holds his 
handle to the bottom of the arrow, an error will be introduced, 
as indicated in Fig. 61. An error will also arise if the follower moves 
the head of an arrow against which he may require to hold. Where, 
owing to slope, &c, the chain handle requires to be held some 
distance above the ground, careful use of the plumb-bob is neces- 
sary if error in marking the true distance on the ground is to be 
avoided. Some error is inevitable if the rough and ready method 
of dropping stones or arrows is employed. 

The error in marking chain lengths may be positive at one chain 
length and negative at another, and on a line of considerable length 
the positive and negative errors will partly balance. The error is 
said to be compensating. 

Errors and Mistakes in Reading the Chain or Tape. — Such errors 
are usually only apt to occur once in measuring a given length, 



CHAIN SURVEYING— ERRORS 65 

namely, in reading the final fraction of a chain length. The 
mistakes liable to occur through reading the chain backwards and 
through reading on the wrong side of the decimal divisions, as 
29 ft. for 31 ft., have been already referred to. In using the cloth 
tape beginners are apt to read 6 for 9 and vice versa if they happen 
to hold the tape with the figures upside down. The same cause 
sometimes leads to 32 being called out where the correct distance 
is 23. To avoid mistakes such as the above one must know where 
they are likely to arise and be constantly on the look out for them. 

Errors in Ranging out Survey Lines. — The effect of any deviation 
of a survey line from the straight likely to occur in ordinary work 
may be entirely neglected as a source of error in the length of the 
line. Careless lining is, however, an important source of error in 
the 'position of details located from a survey fine. A long line 
chained without the use of intermediate ranging poles as guides 
will usually deviate somewhat from a straight fine. The same is 
to be expected with any line ranged out by the eye over a hill or 
across a valley. A survey line though curved on the ground will 
be represented by a straight line on the plan, and hence any objects 
located from the survey line will be out of correct position on the 
plan by the extent of the deviation. In uneven country this forms 
one of the principal sources of inaccuracy in chain surveying and 
is almost unavoidable when the unaided eye is used to range out 
the lines. 

Errors in Locating Objects from Survey Lines. — An error in the 
length of a survey line will affect the accurate plotting of all other 
survey lines subsequently connected to it. An error in an offset 
or tie from a survey line will usually only affect the plotting of one 
point, or at most, a small portion of detail. The important factor 
in determining whether some of the ordinary sources of error in 
measurement need be reckoned with at all in the case of offsets 
and detail measurements, is the scale of the plan. A careful 
draughtsman can plot to T £o m -> so that on the scale of 1 in. = 
100 ft., the smallest distance which can be definitely plotted and 
scaled will be one foot. It would be manifestly absurd in such a 
case to take the detail measurements to the nearest inch. The 
offsets and ties might in practice be taken to the nearest half foot 
and still leave a margin sufficient to cover ordinary errors of 



66 SURVEYING 

measurement, such as inaccuracy of tape, tape not held quite 
horizontal, &c. 

In working to a scale of I in. to 40 ft. it would be appropriate to 
read to the nearest 3 in. or half link. To the scale of 2^oo> which 
is about 1 in. to 208 ft., measurements need only be recorded to the 
nearest foot. The limitations as to the general accuracy attainable 
in chain surveying render futile any attempt to read to closer than 
3 ins. in the detail measurements. 

The foregoing considerations indicate that with ordinary care 
errors in the length of offsets and ties should not be so large as to 
affect the accuracy of the survey. The chief sources of error in 
locating details are to be found in offsets not taken truly perpen- 
dicular, ties forming ill-conditioned triangles, and mistakes in 
reading the tape. 

Mistakes in reading the tape and mistakes in booking the measure- 
ments are the most serious sources of error in locating details. 
Mistakes of the former class have already been referred to under 
errors in measuring a length. Mistakes due to misapprehending 
the figure called out by an assistant will be practically avoided 
if the surveyor repeats the figure aloud. The assistant will then 
correct him if he finds that the figure has been wrongly taken up. 
A surveyor will sometimes write down a wrong figure though he 
has the correct one in his mind. This is rather liable to happen 
if another distance is called out while he is writing the previous one 
down. 

The largeness of the mistakes in the above classes often leads 
to their detection during plotting, but they may often remain 
undetected. Small mistakes of a few feet are most dangerous, as 
they are likely to pass unnoticed and to remain as errors in the 
survey. 

Permissible Error. — The accuracy attainable in chaining is very 
largely dependent on the nature of the ground. Under equal 
conditions the steel band gives much better results than the steel 
chain. An error not exceeding 20011 or 1 ft. in 2,000 ft. may be 
easily attained with a steel band on uniform ground without special 
precautions. It may be very difficult to keep within an error of 
touo" using a steel chain on undulating ground. In the latter case 
it has to be recognised that the attainable accuracy is in large 



CHAIN SURVEYING— ERRORS 67 

measure limited by the imperfections of the human eye as an instru- 
ment for ranging straight lines. There is thus a considerable 
extent of unavoidable error, and it is not profitable to spend much 
time and trouble in eliminating sources of error which are very 
small in comparison. The main endeavour should be directed 
towards attaining the degree of precision required for the purpose 
in hand, and to this end most attention should be paid to the 
eliminating of those sources of error which are relatively largest. 

Expedition in Surveying. — The time taken in plotting a survey 
usually bears no inconsiderable proportion to the time spent in the 
field, and may sometimes be quite as long. Anything, therefore, 
which facilitates and shortens the plotting may as surely further 
the expedition of the work as anything which tends to curtail the 
duration of the field operations. 

For good progress in the field the first essentials are a well- 
arranged system of survey lines and method and order in the field 
operations. Time spent in adjusting the survey fines so as to avoid 
difficult ground and he close to the main portions of the work will 
usually be well repaid. It is in general unwise to attempt to 
locate a large extent of detail from one survey line. If the work 
is complicated time may be wasted in adjusting and altering the 
field sketches, and necessary measurements are apt to be omitted, 
causing much trouble and waste of time. The running of an extra 
survey line at complicated portions so as to bring the fines closer 
to the work and render them less complicated will often be amply 
repaid in the resulting simplicity, clearness and completeness of 
booking and ease of plotting. 

If speed in running the survey lines is to be attained the assistants 
must be made to understand their duties and the system and order 
of operations. In many cases the desired results in surveying, 
as for example in the location of details, can be obtained in several 
different ways. An interesting and instructive survey might be 
carried out in which all the different methods of executing the 
various operations were illustrated, but such a survey would not be 
an expeditious one. Operations to effect a given result should as 
far as possible always be carried out in the same manner, that 
method being chosen which is simplest and best suited to the 
circumstances of the survey. With work thus arranged and 

f2 



68 SURVEYING 

standardised the assistants after a short experience will know without 
instruction from the surveyor what measurements to take and how 
to take them throughout much of the routine work. The surveyor 
must take pains to make his instructions to his assistants definite 
and explicit, especially so in regard to any unfamiliar procedure. 
Instructions wrongly executed through being misunderstood may 
give rise to much annoyance and delay. 

Care and patience are well expended in cultivating and acquiring 
a good style of note-keeping. Badly formed figures, indefiniteness 
in marking the points between which a figured distance extends, 
and crowded and confused notes are inimical to speedy and accurate 
plotting. 



i 



CHAPTER VII 



CHAIN SURVEYING — SPECIAL PROBLEMS 

This chapter deals with the execution of certain operations and 
the solution of some special surveying problems without the use 
of instruments for measuring angles. The operations and problems 
include setting out a right angle, dropping a perpendicular from a 
point on to a chain line, setting out a parallel line, setting out a 
given angle, chaining past an inaccessible area, chaining past 
obstacles, finding distance across a river, surveying far side of river, 
surveying pond, wood, &c, chain surveying in towns. 



Setting out a Right Angle. — Three methods of setting out a right 
angle without the use of any instrument other than the tape or 
chain are shown in Figs. 62, 63, and 64. It is required in each 
"> 




Fig. 62. 





Fig. 64. 



Fig. 63. 



Setting out a Eight Angle. 

case to set out from point B a line perpendicular to the line AB. 
A triangle, the lengths of whose sides are proportional to the 
numbers 3, 4, and 5, is a right-angled triangle, since 5 2 = 3 2 + 4 2 . 
The application of this principle in setting out a right angle is illus- 



70 



SURVEYING 



trated in Figs. 62 and 63. In Fig. 62, AB is measured off equal to 
40 ft. One end of the chain is then held at A and the mark at 
80 ft. is held at B. An assistant then takes the 50 mark and pulls 




20 Loop 



Fig. 65. — Eight Angle by- 
Steel Band. 



Fig. 66. — Perpendicular from a 
Point. 



the chain out taut, thus forming a triangle with sides 40 ft., 50 ft. 

and 30 ft. long. The side 30 ft. long is then at right angles to AB. 

If the chain used is a band it will be better, in order to avoid a 

sharp bend in the chain, to hold one handle at A and the other 

handle at B and then to pull the chain taut while bringing the 

50 mark to coincide with 

the 30 mark, as shown 

in Fig. 65. 

In the method of Fig. 

63, AB and BC are each 

set off equal to 30 ft. 

One end of the 100-ft. 

chain is held at A and 

the other end at B. The 

chain being then pulled 

taut from its centre 

point D, a line joining 

the latter to point B 

will be at right angles 
Fig. 67.-Perpendicular from a point. to AR A ^ ^^ 

perpendicular will be obtained if the above process is repeated to 
find a point E on the other side of AB. The line joining E to D 
will be 80 ft. long and at right angles to AB. 

Another very useful arrangement of lengths forming a right- 
angled triangle is shown in Fig. 64, the sides being 40, 42 and 58 ft. 
long. 




CHAIN SURVEYING— SPECIAL PROBLEMS 71 

Dropping a Perpendicular from a Point on to a Chain Line. — It 

is required to drop a perpendicular from point C (Fig. 66) on to 

the chain line AB. Take a 

length CD on the tape as 

radius, considerably greater 

than the perpendicular length 

CF. Swing this length about 

C as a centre and find the 



points D and E in which it ■ ' a , ,. , ._ .. . T . 

r , ,, , . .. _, , Fig. 68. — Setting out a Parallel Line, 

cuts the chain line. Take 

point F midway between D and E, then CF is perpendicular to AB. 
Another method is illustrated in Fig. 67. Any two convenient 

points G and H are taken on the chain line, one on each side of the 

required perpendicular. 
A tape is stretched from 
G to C and on to H. 
Keeping the tape fixed 
at G and H, the point 
on it which touches C is 
taken hold of and pulled 
over to the other side of 
the chain line to C The 

line joining C to C is at right angles to line AB and cuts it in the 

required point F. 

Setting out a Parallel Line. — A line is to be set out through point C 
(Fig. 68) parallel to the line AB. Drop a perpendicular CD to the 
line AB and measure its 
length. At a convenient 
point E lay off EF perpen- 
dicular to AB and equal in 
length to CD. The line CF 
is parallel to AB. 

If an optical square or a ^j* ^ f* 



t 

1 


, _ M^ 

\ A 

\ / \ 

\ / \ 


G 

Fig. 69.- 


H K L 

—Setting out a Parallel Line. 



other instrument for setting V- —700' -•}£ 

out a right angle is not Fig. 70.— Setting out an Angle, 

available, it will be better Tangent Method. 

to proceed as in Fig. 69. Points G and H are chosen on the chain 
line to give a well-proportioned triangle with point C, and the 



72 



SUEVEYING 



three sides of this triangle are measured. At a suitable distance 
along the line AB set off the equal triangle KML. The line joining 
C and M will be parallel to the line AB. 

Setting out a Given Angle. — With the aid of tables of trigono- 
metrical functions of angles, any angle may be set out with the 

use of the chain alone or with 
the chain and optical square. 

Tangent Method. — In Fig. 
70 it is required to set off 
at point A a line making an 
angle a with the direction 
AB. Measure off a base AC 
any convenient round num- 
ber of feet in length, say 




-Setting out an Angle. 
Sine Method. 



100 ft. At C set off a perpendicular CD, having a length equal to 
100 tan a. The line AD then makes the required angle with AB. 

Sine Method.— In this method, as shown in Fig. 71, the base AC 
is bisected at point H, and the point F is determined, so that 
FH = HC and FC = AC sin a. The direction AF then makes the 
required angle with the direction AB. 

Method by Sine of Half the Angle.— Fig. 72 illustrates this method. 
Point K is set out so that the distance AK is equal to the base AC 
and distance KC is equal to 



2AC 



Direction AK 



then makes the required angle 
with the direction AB. Of 
the three methods given this 
is the only one which will give 
passable results when the 
angle a approaches 90°. The 




- 50' 

Tig. 72.— Setting out an Angle. 

other two methods should not be used for angles which are much 
over 60°. 



Overcoming Obstacles to Chaining. — In chain surveying a condition 
essential to the attainment of accuracy and expedition is that the 
chain lines should be so arranged clear of obstacles and obstructions 
that they can be lined out and measured directly without any 



CHAIN SUEVEYING— SPECIAL PEOBLEMS 



73 



trouble. As, however, the surveyor must be prepared to deal 
with any unavoidable obstacles that may occur, he should make 
himself familiar with the geometrical principles applicable to the 
solution of the following problems connected with the circumventing 
of hindrances in chain surveying: — 




Fig. 73. — Chaining past Pond, &c. 

Chaining past an Inaccessible Area. — Assuming that in this case 
there is no difficulty in ranging out the line on both sides of the 
obstruction the problem may be simply solved in the manner 
illustrated in Fig. 73. From points C and D on the chain fine on 
either side of the obstruction perpendiculars CE and DF of equal 
length are set off to one side of the chain line, so that the line EF 




Fig. 74. — Chaining past and surveying Pond. 

clears the obstacle. The measured length of EF is equal to the 
omitted distance CD on the chain fine. The boundaries of the 
obstruction may be surveyed from the fines CE, EF, and FD, and 
from similar fines set out on the other side of it if required. 

Another method which might be usefully applied where the 
boundaries of the obstructing area have to be surveyed, is illus- 
trated in Fig. 74. The chain line is made to terminate on either 



74 



SURVEYING 



side of the area in points C and D which lie on the sides of a triangle 
GEF conveniently arranged for surveying the area. The length of 
the omitted distance CD can readily be got by plotting the triangle 
GEF on a separate piece of paper and scaling the distance between 




Fig. 75. — Chaining past Pond, &c. 

the points C and D. The length of CD may also be calculated by 
the principles of trigonometry. 

A simple method which might, under suitable circumstances, 
be applied is shown in Fig. 75. From a convenient point E lines 




Fig. 76. — Chaining past Obstruction. 

EC and ED are set off with the optical square or other instrument 
at right angles to each other so as to clear the obstruction, and 
meeting the chain line in points C and D. The lines CE and ED 
having been measured, the length of CD will be got from the equa- 
tion CD 2 = CE 2 + ED 2 . 

Another method is shown in Fig. 76. From points C and D on 



CHAIN SURVEYING— SPECIAL PROBLEMS 



75 



the chain line lines CE and ED are set out and prolonged, EF being 
made equal to CE, and GE equal to ED. The length GF measured 
on the ground is then equal to the distance CD. 

Chaining past an Obstruction which cannot be seen through. — If 

the conditions are such that the line can be correctly ranged out on 




Fig. "77. — Prolonging Chain Line past Obstruction. 

both sides of the obstacle, then any of the four last-mentioned 
methods may be applied to find the distance CD. 

If, however, the line requires to be ranged out from one end and 
produced for some distance past the obstruction, one or other of the 
following methods might be applicable : — 

Method by Parallel Line. — The chain-line having been ranged out 
from the left-hand end up to point C (Fig. 77), set off equal per- 
pendiculars AE and CF 
a considerable distance 
apart. Range out the 
straight line EFGH 
past the obstruction 
and again set off two 
perpendiculars GD and 
HB each equal to CF. 
Points D and B are in 
line with points A and 




Fig. 78.- 



-Prolonging Chain Line past 
Obstruction. 



C, and may be utilised for prolonging 
the survey line towards the right. The measured length of FG is 
equal to the omitted distance CD. 

Method by Four Right Angles. — The optical square may be used 
to lay out successively the lines CE, EF, FD, DB, as shown in 
Fig. 78. Then if FD be made equal to CE, the line DB will He on 
the prolongation of the line AC and the distance EF will be equal 



76 



SURVEYING 



to the distance CD. This is rather a rough and ready method 
especially as regards the correct ranging out of the line AC beyond 
the obstruction. 




Fig. 79. — Prolonging Chain Line past Obstruction. 

Method by Similar Triangles. — In Fig. 79, CE is set off equal to 
and perpendicular to AC. AE is ranged out to G and GB is set out 
perpendicular to AG, GF being made equal to EG, and FB equal 




Fig. 80. Fig. 81. 

Distance across a Eiver. 

to AE. Point D is then determined by the intersecting distances 
FD and DB, which are each made equal to CE. Points D and B 
lie on the survey line and the measured length of EF is equal to 
the length of CD. 



CHAIN SURVEYING—SPECIAL PROBLEMS 



77 




A somewhat similar method can be devised, using equilateral 
triangles. The methods by triangles are not very satisfactory 
owing to the unavoidable shortness of the sides. 

Distance across a River. — The distance across a river which is 
more than one chain length in width, or where the chain cannot be 
passed across, might be determined by the method shown in Fig. 80. 

Points C and D are fixed on the survey line on either side of the 

river. From C a line CE is set out perpendicular to the survey line 

and of a length preferably 

about equal to CD. From 

point E a line EA is set out 

with the optical square or 

other instrument perpendicular 

to the line ED, and point A is 

marked where it meets the 

survey line. The distance AC 

is measured. The required dis- 

CE 2 
tance CD is then equal to — . 

\jA. 

Another method of using 
right-angled triangles is shown 
in Fig. 81. From a suitable 
point E the line EC is set out 
at right angles to the line ED 
to meet the survey line in 
point C. Line EC is then produced to B, BC being made equal to 
CE, and line BA is set out at right angles to BE, meeting the 
survey line in point A. The triangles ABC and CED are similar 
and equal, and the length of AC is, therefore, equal to the length 
of CD. 

Fig. 82 shows still another simple device, by means of which the 
distance CD may be deduced. The line CE is set out at right angles 
to the survey line. From another point A the line AB is set out 
at right angles to the survey line, point B being at the same time 
put into line with points E and D. If AF is equal to CE the triangles 
BFE and ECD are similar, so that :— 





\ £ 




*$ 




\ 


*& 


F 


B 



Fig. 82. — Distance across a Eiver. 



CD 

CE 



= g,orCD = 



CE X EF 
FB * 



78 



SURVEYING 



As the distances to be measured on the ground are AB, AC and 
CE, the above formula becomes : — - 

CE X AC 



CD = 



AB - CE* 



Surveying Far Side of a River. — The principle of the above method 
can, under suitable circumstances, be applied to surveying the far 
side of a river without crossing it. A conspicuous mark, such as a 
boulder, plant, tree trunk, &c, is noted at each salient point of the 
bank, such as A, B and C in the Fig. 83. Using the optical square, 
points a, b, c are found where perpendiculars from A, B and C 
meet the survey line. To find distance ak, a convenient point D 




Fig. 83. — Surveying far Side of Eiver. 

is chosen on the chain line and DE is set out perpendicular to DA, 
point E being at the same time put into line with a and A. The 

Da 2 

length of aA is then equal to - — . 

hid, 

Surveying Pond, Wood, &c. — A pond or a thick wood through 
which fines cannot be sighted must be surveyed by lines run round 
the outside. Wherever practicable, one large triangle should be 
set out enclosing the area, as indicated in Fig. 84, and such other 
subsidiary lines should be run between the sides of the main triangle 
as are required for picking up the boundaries. In plotting the work 
the main triangle is first laid down. The subsidiary fines are then 
joined up and serve as checks on the accuracy of the work. 

Fig. 85 illustrates the kind of arrangement which may be adopted 
where it is impracticable to lay out one encompassing triangle. A 
polygon ABCD is laid out enclosing the area. To enable the sides 



CHAIN SURVEYING— SPECIAL PEOBLEMS 



79 



of the polygon to be plotted in correct relationship to each other 
some of the adjacent sides must be connected by ties or triangles. 
In the figure the relationship of the two sides meeting at A is fixed 
by prolonging those sides and 
forming the triangle A/e. The 
relationship of the sides meet- 
ing at B is fixed by the tie gh, 
which also serves as a survey 
line. Triangles or ties at the 
angles C and D are not required 
for plotting purposes, but are 
desirable as checks on the 
accuracy of the work. The 
fines might be plotted by first 
setting off the base fAgB and 
then plotting the triangles fke and ghB. 




Fig. 84 



-Surveying Wood, 

eA and Bh produced, 
give the directions of the sides AD and BC respectively. The 
lengths of these sides being then laid off to scale, points D and C 
are fixed, and the scaled distance DC should be equal to the 




Pig. 85.— Surveying Pond, &c. 

distance measured on the ground. The triangle at C affords a 
further check on the accuracy of the work. 

It is evident that the triangle A/e might with advantage be 
plotted in the position A/V, and that less error would be introduced 
in plotting if it were laid off considerably magnified. That is, 
instead of plotting the triangle A/V to the actual measured dimen- 



80 



SURVEYING 



sions, plot it to some multiple of those dimensions as A/"e", and 
similarly with the other small triangles. 

Chain Surveying in Towns. — In towns the survey lines must neces- 
sarily for the most part run along the streets, and their arrangement 
into a system of triangles is impossible. Chain surveying is, there- 




FlG. 86. — Chain Surveying in Towns. 

fore, not adapted for town work. A length of street might, however, 
be quite well taken up with the chain alone along with short lengths 
of the branch streets. The branch survey lines are connected to the 
main survey line by triangles at the junctions of the streets, and 
as the accuracy of the work depends mainly on these triangles 
they must be made as large as practicable. Fig. 86 illustrates the 
sort of arrangement which may be adopted. 



CHAPTER VIII 

PLOTTING THE PLAN 

This chapter deals with the drawing instruments used in the 
plotting of survey plans and with the methods of procedure applic- 
able to the commencing of the plan and its arrangement on the 
sheet, the laying off of the base line, the plotting of the main 
triangles, and the plotting of offsets, ties and details. Consideration 
is given to the errors which are apt to arise in plotting, and hints are 
given as to the proper methods of pencilling, inking-in, and erasing 
lines. Conventional methods of representing various features, 
such as fences and boundaries, roads, railways, buildings and 
various kinds of land, are considered and illustrated, and the 
matters of printing and lettering, colouring and tinting are also 
dealt with. 

Paper. — For plans intended to remain as a permanent and accu- 
rate record of the work surveyed good, tough, seasoned hand-made 
paper should be used. The surface should be moderately rough 
and the quality such that ordinary rubbing and erasures can be 
made without spoiling the paper for taking colour and ink lines. 
Whatman's hand-made paper, double elephant size, is suitable for 
single plans, which should preferably be kept flat in a drawer. 
When required to stand much handling the paper should be mounted 
on holland. For large plans several sheets of paper may be mounted 
together on cloth. Plans which extend over a long narrow area, 
such as the general working plan of a railway, are often made on a 
continuous roll formed of single sheets mounted together on cloth. 
Rolled plans do not remain so true to scale as plans which are kept 
flat. 

Pencils. — A good quality pencil, hardness 3H or 4H, should be 
used for plotting. For marking points and distances a hard pencil 
with a very fine round point may be used, but it is preferable to 



82 SURVEYING 

employ a pricker with needle point for this purpose. For drawing 
lines a fine chisel point should be used. The pencil should be held 
nearly vertical, its top being just slightly inclined forward in the 
direction of motion, and the flat side of the point held close against 
the straight-edge or ruler. For a right-handed person the directions 
of motion in drawing lines should be from the left-hand side of the 
paper towards the right-hand side and from the bottom towards 
the top. 

Pricker. — A serviceable pricker may be made by fixing a portion 
of the point end of a sewing needle into the end of a pencil. The 
leg of a pair of dividers or compasses, if furnished with a good point, 
will often serve the purpose. Points marked off with a pricker may 
be rendered conspicuous, so as to be readily recoverable when wanted, 
by drawing a small pencil circle round them. 

Scales. — Surveys made with the 66-ft. chain are generally plotted 
to one of the following scales : 10, 20, 30, 40, 50, 60, or 80 links to 
1 in., or to some round multiple of one of those scales. If distances 
are required to be scaled in feet on a plan plotted to one of the 
above link scales, either a specially manufactured scale must be 
used, or a scale of feet corresponding to the link scale must be 
drawn. A scale of 66 ft. to 1 in. corresponds to the scale of 100 links 
to the inch. The foot scales corresponding to the ordinary link 
scales will, therefore, always contain an awkward number of feet 
per inch, namely, some multiple or sub-division of 66. 

Surveys for engineering purposes are usually made with the 
100-ft. chain or steel band, and plotted to one of the following 
scales or some round multiple of one of them : 10, 20, 30, 40, 50, 60, 
or 80 ft. to 1 in. In connection with railway work in Britain, 
survey plans are commonly plotted in feet to the scales 1 in. = 
33 ft., and 1 in. = 66 ft. 

The scales 4, 8, 16, 32, and 64 ft. to the inch are much used by 
architects. On plans drawn to these scales distances can be 
measured roughly with an ordinary foot-rule. 

The principal scales of the Ordnance Survey maps of Britain are 
the following : the ^ ^l^, and 6 in. to a mile or in ^. The ^ 
is known as the large Ordnance Scale and is used for the survey 
plans of towns. Maps of all cultivated portions of the country are 



PLOTTING THE PLAN 



83 



prepared to the 2 - s 1 tfg scale, which is equivalent to 208^ ft. to the 
inch, and maps of the whole country are prepared on the 6 ins. to 
mile scale. 

In plotting the plans flat boxwood rules about 12 ins. long, with 
both edges bevelled and a different scale engraved on each are 
most commonly used. The divisions are figured decimally. Scales 
whose divisions on one edge are intended to represent links may 
have the other edge divided into feet to correspond. A common 
form of scale, as shown in Fig. 99, has 10 divisions to the inch along 
one edge and 20 along the other, other similar combinations being 
30 and 40 divisions to the 
inch and 50 and 60 divisions 
to the inch. 

The main divisions are 
numbered from both ends so 
that distances may be scaled 
and marked off in both direc- 
tions. The scale marked 10 
may be used to plot work to 
the scale of 1 in. to 1 ft. In 
that case the numbers along 
the scale will be read as feet, 
and each small division will 
represent ^ of a foot. Or it 
may be used to plot to the 
scales of 10 ft., 100 ft. or 
1,000 ft. to the inch. In the 



A 





Fig. 87. — Set-squares. 



same way the scale marked 20 can be used for plotting to 2, 20, 
200, «fec, feet to the inch. 



Set-squares. — The draughtsman should have two good-sized 
triangular set -squares for drawing perpendicular and parallel lines 
and for general use as straight-edges in drawing and inking-in short 
lines. Fig. 87 shows common forms of set-squares and sizes suitable 
for general use. Triangular framed set-squares of pear wood or 
mahogany with ebony edge are reliable and cleanly in use. Set- 
squares formed of triangular sheets of vulcanite or transparent 
celluloid are now common. They pick up dirt and soil the paper 
much more readily than wooden set-squares and should always be 

G 2 



84 SURVEYING 

rubbed clean before use. It is essential that the set-squares used 
in plotting survey plans should be accurate, especially as regards 
the right angle. To test if this is correct apply the base of the set- 
square to a straight-edge on a sheet of paper, as shown in Fig. 88, 
draw a pencil line along the perpendicular side, and reverse the set- 
square. If the edge coincides 
with the pencil line the right 
angle is correct. 

Straight-edge. — All instru- 
ments employed for drawing 
accurate straight lines must 

have their edges perfectly true 
Fig. 88.— Testing Set-square. . Jf, 

and straight. The accuracy 

of a drawing edge may be tested in the manner illustrated in 
Fig. 89, by drawing a pencil line and then reversing the straight- 
edge, end for end. 

Long straight-edges are most reliable if made of steel. A suit- 
able size for drawing long base lines, &c, is 6 ft. in length and about 
2*5 ins. by 0*1 in. in cross-section with one edge bevelled. 

If accurate straight lines are required longer than the length 
of the available straight-edge, a fine thread should be stretched 
tight between the ends of the line and a few points along its length 




Fig. 89.— Testing Straight-edge. 

transferred accurately to the paper by means of a pricker. The 
straight-edge should then be used to join up the line through these 
points. 

Compasses. — The various instruments used in drawing circular 
arcs are spring bows, bow compasses with or without lengthening 
bar, beam compasses, and manufactured curves. These are illus- 
trated in Figs. 90 to 95. 

Spring bows can be accurately adjusted by the thumb screw to 
the exact radius required, and are used for drawing small circles 
within about an inch radius. Bow compasses with jointed legs 



PLOTTING THE PLAN 85 

are used for circles up to about 5 ins. radius, and with lengthening 
bar up to about 9 ins. radius. In using the compasses the legs 
should be bent at the joint so as to bring the points perpendicular 
to the paper, this being specially necessary in the case of the drawing 
pen point. Compasses used with a long lengthening bar are apt to 
be deficient in steadiness. 

In the beam compasses, as shown in Fig. 94, the pivot point and 
the drawing point form detached portions of the instrument and 
can be clamped at any required distance apart on a beam of wood 
which, for the sake of stiffness, is usually made of T section. The 
drawing point is furnished with a tangent screw by means of which 
the final adjustment to the exact radius is made. To set the beam 
compasses to the required radius, first mark off by scale the length 
of the radius along a pencil line drawn, say, near the edge of the 
sheet of paper. Then, having set the points approximately to this 
length, apply the pivot point to one end mark and turn the tangent 
screw until the drawing point coincides with the mark at the other 
extremity of the length. The pivot point is then set to its centre 
and the arc is struck. 

For drawing accurate pencil circles of large radius a narrow strip 
of tough seasoned drawing paper serves excellently in place of the 
beam compasses. The required radius is marked off along a pencil 
fine drawn down the centre of the strip, and fine holes are pricked 
through the end points. A pricker or common pin passed through 
one of these holes forms the pivot, and a fine pencil point passed 
through the other serves to draw the arc. Just sufficient tension 
should be maintained on the paper strip to keep it taut. 

Manufactured Curves, formed of flat strips of pearwood, card- 
board or vulcanite, and having both the convex and concave edges 
turned to the same radius, can be obtained in sets having radii 
varying from 1| ins. to 240 ins. or more. These are useful for 
drawing and inking-in curved fines, and are indispensable for 
railway work. 

Dividers. — The dividers, similar to the bow compasses, but with 
both legs stiff and sharp-pointed, are not much required in plotting 
plans. Their chief function in surveying work is to measure dis- 
tances on the finished plans. Short distances are measured by 
setting the dividers to the whole length and then applying them to 



86 SURVEYING 

the scale on the plan. Longer distances are measured by stepping. 
The dividers are set to a round distance on the scale, and then 





Fig. 90.— Pen Spring Fig. 91.— Pencil 
Bows. Spring Bows. 





Fig. 92.— Pen and Pencil 
Bow Compasses. 



Fig. 94. — Beam Compasses. 





Fig. 95. — Manufactured Curves. 



Fig. 93.— Dividers. 



stepped along the line from one end, the equal distances thus 
measured off being added up mentally as the stepping proceeds. 
The last fractional interval is taken on the dividers, measured on 



PLOTTING THE PLAN 87 

the scale, and added to the total length of the equal steps to give the 
total distance between the points. 

For use in the field the surveyor should be provided with a pair 
of pocket dividers, having a screw-on shield or other device to pro- 
tect the points. Ordinary dividers may be carried in the pocket 
if their points are inserted in a piece of cork or indiarubber. 

Parallel Ruler. — The ordinary T-square sliding on the edge of a 
drawing board is of little service in plotting surveys. A long, 
heavy, parallel ruler is more generally useful. The instrument, 
which should be at least 2 ft. long, consists of a heavy bar of brass 
or electrum about 2| ins. wide, with parallel bevelled edges and 
mounted on a pair of rollers (Fig. 96). The rollers are of equal 
diameter and are rigidly connected to the same axle so that they 
roll together and travel over equal distances. The ruler, therefore, 
keeps a parallel direction as it travels across the paper. To test 



Fig. 96.— Parallel Euler. 

the accuracy of a parallel ruler, draw a pencil line along its edge 
on a sheet of paper, then roll the instrument a considerable distance 
across the paper and draw another line along the same edge. Re- 
verse the ruler end for end, apply the same edge to the first line, 
roll the instrument across the paper and see if the edge now coin- 
cides with the second drawn pencil line. If it does the ruler is 
correct. 

With the use of the parallel ruler and set-squares parallel lines 
and perpendicular lines can be drawn in any direction on the paper. 

Drawing Pen. — The drawing pen is illustrated in Fig. 97. The 
nibs should be of good quality tempered steel, otherwise they soon 
become blunt and sharp uniform lines cannot then be drawn. The 
pen is more easily cleaned, sharpened, and adjusted if one of the 
nibs is hinged so as to open wide when the screw is taken out. To 
get good fines the nibs must be sharp, of the same length and shape, 
and exactly opposite one another. In drawing ink lines the pen 
should be held and handled in a similar manner to the pencil, with 



88 SURVEYING 

the screw head away from the edge of the ruler. Uniform firm 
pressure should be exerted downwards on the paper, and only a 
very slight constant pressure against the straight-edge, so as to 
ensure lines of even thickness. Ink should not be allowed to dry 
between the nibs. They should be cleaned out thoroughly and 
often by drawing the fold of a duster between them without altering 
the screw. 

Commencing the Plan. — The scale of the plan will be fixed from 
considerations of the purpose for which the plan is intended, and 
the size of the sheet of paper required will be governed 
A by the scale chosen. Having determined the necessary 
size of the sheet, lay it down flat on a table or board, 
previously rubbed clean, and fasten the edges down 
with cloth- or leather-covered weights, or with drawing 
pins. See that the instruments are clean. Keep covered 
over with paper any portions of the plan that are not 
being worked upon. 

Arrangement of Plan on the Sheet. — If an existing plan 
of the area is available, even though to a considerably 
smaller scale, a proper arrangement of the plan on the 
paper can be readily arrived at. Lay off on this plan 
the position of one of the main survey lines near the 
centre of the work, and on a piece of tracing paper draw 
a rectangle representing the size of the sheet on which 
Fig. 97. — the work is to be plotted, reduced to the scale of the 
PeQ Wmg existing plan. Place the tracing paper over the area and 
twist it about until the boundaries of the area appear 
in good position, relative to the rectangle, keeping in view the 
provision of space for the title, scale, north point, notes, &c, 
and the effect these will have on the symmetry of the completed 
plan. Having fixed the rectangle about the area, mark on 
it the position of the main survey fine and its end points, and 
produce this fine both ways to meet the edge of the rectangle. 
Reproduce this line and the survey line in accurate relative position 
on the sheet of drawing paper. This survey line so fixed will serve 
as a base on which to construct the system of triangles, and the 
work when plotted will He in its predetermined position on the 
paper. 



PLOTTING THE PLAN 



89 



Where there is no existing plan available the main triangles 
should be roughly plotted to the scale of the plan and the limits 
of the area sketched round them. The tracing paper can then be 
applied direct over the sheet of drawing paper and, when the area 
is judged to be in proper position, one of the main survey lines may 
be pricked through, measured off accurately to length and used as 
a base for the system of triangles. 



Laying off the Base Line. — A reliable straight-edge must be used 
for laying off the base line and for drawing all survey lines. If any 
line is very long a stretched thread should be used in the manner 
already explained. If it will not be objectionable on the finished 
plan, the base and survey fines may be inked in with thin faint lines 
of blue or red colour. 

A fine pricker will be used for marking off the lengths of lines. 
To ensure accurate work the eye should be placed opposite the 



G 



a 



c 



□ 



J*-.-,- 



Fm. 



-Laying ofi long Distance. 



division of the scale to be pricked off, and the pricker should be held 
vertical. Long distances will be marked off in whole scale lengths. 
or the nearest round figure if the scale ends at an odd distance. 
The most accurate method of laying off long distances is by using 
two scales in the manner indicated in Fig. 98. The scales are 
placed one on each side of the line, their end divisions are made to 
coincide, and they are alternately moved forward and fixed down by 
weights. 

Plotting the Main Triangles. — As a preliminary to the plotting 
of the main triangles the surveyor should prepare a sketch of the 
system of survey lines with all stations numbered, and with the 
lengths of all sides of triangles and other survey fines plainly 
figured. By keeping this before him during plotting he will save 
much turning up and searching of the notebook. 

The triangles which are directly connected to the main base 
line will be plotted first. To plot the third station of a triangle, 



90 SURVEYING 

from the ends of the base as centres sweep intersecting arcs with 
the lengths of the sides as radii. The intersection of the arcs fixes 
the position of the station. Where the lengths of the sides are 
within the length of the scale it is quicker to draw only the first 
arc for each triangle and to use the scale itself as radius for finding 
the intersection instead of the second arc. Before permanently 
marking a station found thus by intersections test whether both 
sides of the triangle scale correctly. If there is a slight error 
adjust the point to true position. Then test its accuracy further 
by any proof lines which may have been taken. If the result is 
satisfactory, make a definite prick mark at the point and draw in 
the sides of the triangle. Proceed similarly with the rest of the 
triangles attached to the base line and then with those more remote. 
No plotting of detail should be commenced until the whole system 
of triangles has been completed and checked. 

Plotting Offsets. — The plotting of offsets and ties will follow 
roughly the order in which they were measured in the field. Where 
few offsets occur on a survey fine the quickest method will be to 
first mark off along the fine the positions of all offset points, and 
then erect perpendiculars at these points and prick off the lengths 
of the offsets to scale. Offsets which have been set out at right 
angles by the eye in the field may be plotted by placing the 
scale at right angles by estimation across the survey fine. 
Perpendicularity of the longer offsets should be ensured by 
using the right-angle set-square. 

Where there are many offsets the method of plotting illustrated in 
Fig. 99 is recommended. A short sliding offset scale is used, having 
the zero of graduation at its middle and with an index mark on the 
sliding base. The scales are set ready for use by placing the offset 
scale at right angles to a survey fine with its zero at the commencing 
station and then setting and fixing the ordinary scale parallel to 
the survey fine and with its zero at the index of the offset scale. 
On sliding the offset scale its zero should travel along the survey 
line. Then to plot, say, an offset of 27 ft. at distance 253 on the 
survey fine, set the index to 253 on the ordinary scale and make a 
prick mark at division 27 on the offset scale. By this method of 
plotting, points do not require to be marked off along the survey 
line, and the drawing of pencil perpendicular fines is avoided. If 



PLOTTING THE PLAN 



91 



a continuous boundary is being plotted the points should be joined 
up freehand as they are plotted. Isolated points which are not 
immediately required should be marked with a pencil circle for the 
sake of easy recovery. 

Plotting Ties. — Ties are plotted by intersecting arcs drawn with 
the spring-bow compasses. Where the offset scale is being used to 
plot perpendicular offsets it may also be used to mark off the centres 
for the ties along the survey line. Wherever check measurements 
have been made they should be applied to test the accuracy of the 
work as the plotting proceeds. 

Plotting Details. — Details of houses and other objects which have 
been separately sketched should be plotted concurrently with the 




Fig. 99.— Plotting Offsets, &c. 

survey lines. Errors are more readily avoided and sooner detected 
if all the plotting is completed as the plan progresses. 

Errors in Plotting. — The error in measuring off a distance with an 
engine-divided scale should be very small. It should be within 
xoo m - m an y sm gl e measurement, and should not much exceed 
that amount in the length of a double elephant sheet. The serious 
errors in plotting arise principally from misreading the notebook, 
blunders in reading the scale, measuring from the wrong point, 
drawing lines between the wrong points. The notebook should be 
so placed in front of the draughtsman that figures and writing can 
be read in their natural position. Errors in plotting due to mis- 
reading the scale usually amount to an even round number of feet. 
An error of 10 ft. is probably most common, and is most liable to 



92 SURVEYING 

occur when the scale is figured only at alternate divisions, namely, 
the even tens. Mistakes of 1 ft. and 100 ft. also occur. Large mis- 
takes of this class will generally be detected. With scales whose 
divisions are figured from both ends the mistake may be made of 
looking at the number on the wrong row, and hence marking off 
the wrong distance. This mistake is most liable to occur at dis- 
tances near the centre of the scale. Errors due to measuring from 
the wrong point, such as laying off the wrong offset from a point on 
a survey fine, or errors due to drawing lines between the wrong 
prick marks can be avoided only by exercising great care and 
constantly applying checks. Serious errors are, as a rule, oftener 
introduced in the plotting than in the field work, and the employ- 
ment of systematic methods of procedure in plotting and checking 
is essential to the attainment of an accurate plan. 

Pencilling. — The pencilling of the work should be done in fine, 
firm fines. Do not press so hard on the point as to indent the 
paper or render the fines difficult to erase. Particular care must be 
taken to see that the proper prick marks are joined up. In pen- 
cilling boundary lines pay attention to the definite angles and see 
that the lines are drawn through all the plotted points. So far as 
possible leave the prick marks visible so that the ink fines may be 
drawn exactly through them. In plans of single sheet size com- 
plete the pencilling and check it carefully before starting to draw 
the lines in ink. With large plans or long rolled plans it may be 
advisable to do the inking-in in sections as the plotting proceeds, 
as pencil lines soon become blurred and indistinct when a plan is 
subjected to much handling. 

Inking-in. — Chinese ink freshly rubbed down from the stick 
gives best results. See that the porcelain dish is clean before 
rubbing down, and keep the ink covered to prevent the access of 
dirt and retard evaporation. Make the ink thick enough to give 
a dense black line, but not so thick that it clogs the pen. The 
lines should not be drawn too narrow. They should be wide 
enough and firm enough to remain distinct after a plan has become 
rubbed and dirtied by much use. If satisfactory permanent ink 
lines are desired, it is essential that the drawing pen should be sharp 
and pressed firmly into the paper. Lines drawn with a blunt pen 
are easily rubbed off. If a line as first drawn appears ragged or 



PLOTTING THE PLAN 93 

contains gaps owing to the paper being greasy in places, go over 
the line again in the same direction with the drawing pen, taking 
care not to make it too thick. As a general rule, it is desirable to 
commence inking at the top left-hand corner of the paper and to 
work downwards and to the right, and curved lines, especially if 
drawn with a compass, should be inked in before the adjoining 
straight lines, the positions of the tangent points having previously 
been carefully marked in pencil. Where, however, manufactured 
curves are used for ruling in the curved lines, as in drawing railway 
lines, &c, it is better to work continuously in one direction (left to 
right), taking the straights and curves in order. In pencilling 
railway curves and such like on a plan, jot down the radius of curve 
used and mark its tangent points. Draw the ink lines accurately 
over the prick marks so as to hide them if possible, see that the lines 
meet exactly at angles, without overlapping or showing any gap, 
and take precautions to avoid the sleeve, ruler, or set-square rubbing 
over the wet lines. Keep the plan as clean as possible, by covering 
up the portions that are not being worked upon, and see that 
there is no dust or eraser rubbings in the path of the drawing pen. 

Erasing. — Erasures of ink lines are difficult to effect and always 
leave their mark on the surface of the paper. The use of a knife 
for erasing lines usually results in spoiling the surface. Good ink- 
eraser — a hard rubber mixed with grit — applied gently and with 
patience is best for the purpose. When the line has been erased 
brush off the dust and grit, rub the erasure over with soft india- 
rubber to remove any grit sticking to the paper, clean away the 
effects of this rubbing, and then polish the surface of the paper 
with some hard and smooth substance, such as the end of an ivory 
scale or the rounded end of an ivory drawing pen handle. Fine 
sandpaper is sometimes useful for erasing purposes where the ink 
lines have been well cut into the paper. 

Conventional Signs. — Some of the conventional signs employed 
in representing various objects and features on survey plans are 
illustrated in Figs. 100 to 109. 

Full Black Lines. — All definite and permanently marked boun- 
daries and outlines of existing objects should in general be repre- 
sented by full black lines. 



94 SURVEYING 

Dotted Lines. — Dotted lines are employed to represent boundaries 
which are indefinite and generally somewhat unimportant. Paths, 
unfenced roadways, edges of slopes, unfenced edges of woods, kerb 
lines of streets, and divisions between cultivated land and moorland 
are examples of features which are generally shown by dotted lines. 
Dotted lines appear neatest when drawn with very short uniform 
dashes, almost dots, closely and uniformly spaced. 

Coloured Lines. — Coloured inks and water colours are not so 
permanent as Chinese black ink, and some colours fade rapidly. 
They should be used as sparingly as possible on plans intended to 
be kept as permanent records. Proposed works, proposed divisions 
of land, &c, are usually drawn in red lines. Railway lines are 
often drawn in blue. 

Fences and Boundaries. — On small scale plans all sorts of definite 
boundaries, such as fences, hedges and walls, are usually represented 
by a single full black fine. See Fig. 100. 

On plans drawn to a fairly large scale, a dot-and-dash line is 
generally used to distinguish fences from other boundaries. A 
good appearance is obtained if the dashes are made about 0-15 in. 
long, uniformly spaced, and with intermediate dots which are 
merely round dots and not short dashes. 

Walls are shown by double full lines when the scale is large 
enough to show the thickness. 

Hedges are shown by single full lines with a conventional repre- 
sentation of bushes drawn over them. 

Some conventional representations of gates are shown on the 
boundaries illustrated in Fig. 100. 

Roads. — The methods of representing roads vary with the scale 
of the plan, and are generally as indicated in Fig. 101. 

Railways. — The methods of representing railways vary also 
with the scale of the plan. Usual conventions to represent a 
single-line railway and a double-line railway where the scale is too 
small to permit of showing the actual width of the railway or details 
of cuttings and embankments are shown in Fig. 102, where also 
a portion of a double-line railway is shown, with the amount of 
detail appropriate to a scale of 1 in. to 80 ft. or larger. 



PLOTTING THE PLAN 



95 



FENCES AND BOUNDARIES, 

Fence, Hedge or Wall (Small Scale) .^ 



Hedge and Gate 
Fence and Gate 
Wall and Gate 

Path (Small Scale) 
Path or Road unfenced 
Road Fenced 

Road (Large Scale) 



tXJ: 



F'S 100. 
ROADS 



T 



^9 101. ' 

R A I L WAYS. 
Single Line (Small Scale) . v**' 1 *'*""""'"*^^ 

Double Line (Small Scale) 



H^ 



Railway, Cutting, 
Banking & Bridge 



sniHRiffl 



"fig. 1-02. 



Example from Survey Pla 



PARISH 



N E. I L S T O N 




Eyat Limt Reservoii 



96 



SURVEYING 



BUILDINGS 



il|j|L_ 





Shed 
House. with open side. Glasshouse 

Fig. 103 . 



l/ARIOUS KINDS OF LAND 



- 4 



Fi g . 104. Fir Wood 






L s%£ 



4 



4* *s. ^5L 



'» _ 



Fig 105. Mixed Wood. 



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C£ C% 


djjk 


ej& 


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4 4. 


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Fiq.l06.0rchard. 





M1IIUM, 




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Fig 107. Moorland 




Fig. 108. Marsh and Pond 




Fig.l09.Cliffs and Shore 



PLOTTING THE PLAN 97 

Buildings. — Methods of representing buildings are shown in 
Fig. 103. Sheds with open sides are shown dotted in outline. 
Glass-covered houses are generally shown cross-hatched in black 
or blue lines. Masonry buildings may be distinguished from wooden 
buildings by showing the former coloured in light Indian ink and 
the latter in light brown, but it is quite common to colour all 
buildings light Indian ink. An effective method of showing up 
buildings is to hatch them with thin parallel ink lines evenly spaced. 
This method is more laborious than colouring, but causes less dis- 
tortion of the plan. The employment of shade lines helps to make 
the buildings stand out more graphically. To fix which lines should 
be emphasised by thickening, imagine that rays of light are coming 
from the top left-hand corner of the plan and crossing it diagonally 
at 45°. Those sides which the light strikes remain of ordinary 
width, the others, which would be in shadow, are thickened towards 
the inside of the building. Shade lines may, with good effect, be 
about three times as thick as the ordinary lines. 

Various Kinds of Land. — Conventional representations of fir wood, 
mixed wood, orchard, moorland, marsh and pond, and rocky shore 
with high and low water marks are shown in Figs. 104 to 109. 

Example from Survey Plan. — Plate I., which is copied from a 
portion of a survey plan of a double-line railway plotted to the 
scale of 1 in. to 66 ft., and reproduced here to a somewhat smaller 
scale, embodies a considerable number of conventional signs and 
gives a good idea of how they should be applied on a survey plan. 
On the survey plan from which the plate is copied the rails were 
shown in blue lines, the water of the reservoirs was shown by 
shading in blue colour round the edges, and the shade lines of the 
earth slopes were drawn in thin Indian ink, and were consequently 
less pronounced than the full black lines of the plate. 

The plate also shows scale, north point, and specimens of lettering. 

Printing and Lettering. — Plate II. shows the usual styles of 
lettering which may be employed in printing the necessary informa- 
tion on a survey plan. 

The student who desires to become proficient at printing and 
lettering should start with the practice of the vertical block printing. 
His aim in the first place should be directed towards the attainment 



98 SURVEYING 

of correct form and proportion in making the letters, and he must 
not be satisfied till he can form the letters directly with the printing 
pen without any preliminary pencilling. The acquirement of a 
rapid, neat and effective style of finishing the letters will then come 
as the result of further and persistent practice. 

When the student has mastered the vertical block printing he 
will find that the attainment of a neat hand in any of the other 
styles will be a comparatively easy matter. 

The vertical block lettering, varied in size according to the import- 
ance of the particulars described, may be used throughout for the 
lettering of survey plans and, when neatly executed and arranged, 
is always effective and in good taste, but to most draughtsmen it is 
rather a laborious method. The small thick-and-thin italics can be 
rapidly executed and is usually very neat on paper plans, but is not 
very suitable for tracings which are to be used in making sun-print 
copies, as the thin portions of the letters hardly come out on the 
prints. For this reason the small sloping italics of uniform thickness 
is in much more common use nowadays. It is also probably the 
most rapid of all styles of hand printing. 

In order to avoid an appearance of sameness over the plan, it is 
desirable that a limited number of variations of the character of the 
printing should be made in such a way as to emphasise the different 
character and importance of the information which it denotes. 
For example, vertical block letters, Roman capitals, and small 
Roman letters may be used for various classes of important informa- 
tion, while the general run of small printing is done in small italics. 
A study of the published plans of the Ordnance Survey of Britain 
will furnish useful ideas as to the size and character of printing 
appropriate to the various classes of information on the several scales. 

The title of the plan should be formed, as a rule, of simple plain 
lettering in several lines, forming a well-balanced whole, in the form 
of an oval, if possible. Vertical printing has always a much better 
appearance in a title than sloping printing, and for a simple title 
there is probably nothing better than a combination of one or two 
lines of Roman capitals with one or two lines of vertical block, the 
size of the lettering in the several lines being varied with their 
importance. A thin ink fine drawn under a fine of printing will 
often help to make it stand out and appear clean and straight, but 
the overloading of a title with lines and scrolls should be avoided. 



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PLOTTING THE PLAN 99 

The title should generally be made to include : — 

(a) The description or name of the area surveyed or the name of 
the estate, owner or principal for whom the work is done. 

(b) General description of locality and purpose of survey. 

(c) Particular description of the work shown on the plan, usually 
in the form " Plan showing proposed Road," or as the case may 
require. 

The title should where possible be placed above the work on 
the vertical centre line of the sheet, but on survey plans of irregular 
areas the appearance of the plan may often be enhanced by suitably 
disposing the title in some vacant space between the work and the 
edge of the sheet. The title should never be placed within the work 
of the plan. Where the survey occupies a series of plans it will 
usually be preferable to make the title small and place it in the 
upper or lower right-hand corner along with the sheet number, 
so that when a number of plans are lying flat together in a drawer 
any particular one required can be found at once by turning up the 



Colouring and Tinting. — The application of large areas of colour 
to original copies of important survey plans is not desirable, as it 
causes alterations of the dimensions of the paper. On such plans 
where different areas have to be distinguished from one another it is 
better to confine the colour to an edging round the boundary of the 
area. For a small area the edging should be a single narrow band 
of fairly bright colour around the inside of the boundary. The 
brush should be held with its point towards the inner edge of the 
band and applied in short strokes towards the draughtsman. For 
large areas a broader edging consisting of a strip of light colour, 
I in. to £ in. wide, with a band of deeper colour about one-third as 
wide next the boundary may be used. 

In tinting large areas the surface of the paper should first be 
cleaned, and all pencil lines, except those which are left to be after- 
wards inked in, should be erased, as otherwise they will be fixed 
by the colouring. The colour, palette, and water used should be 
clean, and sufficient tint should be mixed down to cover the whole 
area at one application. The sheet of paper should be fastened 
down to a drawing board, the back of which should be raised so as 
to give the paper a slope towards the draughtsman. A large brush 

h2 



100 SURVEYING 

should be used, and the tinting should commence from the top left- 
hand portion of the area and be continued simultaneously towards 
the right and downwards. The wet brush is applied from left to 
right along the upper portion of the boundary and then, with rapid 
motions of the brush, the tinted area is extended downwards. The 
colour drains towards the bottom edge of the wetted portion, and 
the whole art of tinting consists in constantly applying just so much 
colour as will keep the wetted surface continually draining down- 
wards without allowing any large quantity to collect and run off in 
a trickle, and in executing the work so rapidly that no portion of 
the colour along the lower edge has time to dry before it is led 
further down. When the lower boundary is approached the brush 
is used somewhat dry, being freed from colour as required, and is 
employed to collect the surplus tint and complete the colouring to 
the edge of the area, working again from left to right. 

Where overlapping areas have to be distinguished or where an 
area included within a larger coloured area has to be marked for 
separate recognition it is better to use cross-hatching with coloured 
lines for the purpose, rather than to superimpose one tint on the top 
of another of a different colour. 

Date and Particulars of Survey. — An original survey plan should 
always have a note giving the date when the work was executed 
and should be signed by the surveyor in charge. If a plan is 
partly compiled from other plans the note should give particulars 
as to the extent plotted from actual survey, and the extent and 
source of the compiled portion. 



CHAPTER IX 

COMPASS AND SEXTANT SUKVEYING 

The instruments principally used by the surveyor for the measure- 
ment of angles are the surveyor's compass, the prismatic compass, 
the box sextant, and the theodolite. The three first-mentioned 
instruments are used for measuring horizontal angles (the sextant 
can, however, measure angles in any plane), and are light, easily 
portable, and rapid in use, and give results to a useful degree of 
accuracy for certain purposes. The theodolite may vary from a 
light instrument easily carried by one person up to a heavy instru- 
ment requiring special arrangements for its transport, and may 
have widely varying degrees of precision, but contrasted generally 
with the other three instruments, it is much heavier, much more 
awkward to carry, and requires a considerable expenditure of time 
and care in order to set it up on the ground in readiness for reading 
an angle. On the other hand, the theodolite gives much more 
accurate results than the other instruments, and can also be used 
for measuring vertical angles. 

This chapter will deal mainly with the principles, construction, 
and use of the surveyor's compass, prismatic compass, and box 
sextant, and their employment in surveying. 



The Magnetic Needle. — The earth acts as an immense magnet 
and exerts a directive influence on a magnetised bar of cast steel. 
Suppose that a bar of steel symmetrical about its centre is 
not magnetised and is suspended at its centre of gravity 
in such a way as to be free to rotate both vertically and horizontally 
and so that when pointed in any direction it does not tend to move 
away from that direction. Let the bar now be magnetised and 
again suspended as before. It will now oscillate about the point of 
suspension and will gradually settle into a definite position which 
will not be in a horizontal plane. The bar will be pointing in a 



102 SURVEYING 

slanting direction towards the ground. By suitably weighting 
the high end the bar may be brought to a horizontal position, and 
it will then have a definite direction in the horizontal plane, and if 
disturbed it will oscillate about and finally settle back into the 
same position. The horizontal angle which the direction of the 

bar now makes with the direc- 
tion of true north is the 
« " magnetic declination " at the 

= place, and the vertical angle 
Fig. llO.-Broad Needle. through which ^ bar required 

to be rotated to bring it from its o r iginal slanting position into the 
horizontal position is the " magnetic dip " at the place. 
| The two most common forms of the magnetic needle used in 
surveying instruments are the " broad needle," illustrated in 
Fig. 110, and the " edge bar needle," shown in Fig. 111. The 
needles are bored in the centre and fitted with a cap containing 
a hollow coned bearing of agate or other very hard stone. The 
needle swings on a pointed pivot of hardened steel. A lever 
arrangement is usually provided for lifting the needle off its bearing 
when not in use, so as to pre- 



vent unnecessary wear of the 0= a=== i * 

bearing with consequent in- 

crease of friction. The tendency I I tl [] • | | 

to dip is usually counteracted Fia# m._Edge Bar Needle. 

by placing a small sliding 

weight on one arm of the needle. As the magnetic dip is variable 

over the earth's surface the weight may require shifting for different 

localities in order to obtain horizontal balancing of the needle. 

Magnetic North and Magnetic Meridian. — A freely swinging mag- 
netic needle properly balanced and centred and uninfluenced by 
sources of local attraction will come to rest in a direction which 
for most parts of the earth's surface is not widely different from 
true north and south. The end which points in the northerly 
direction is called the " north " end, or better, the " north-seeking " 
end of the needle. The direction in which this end of the needle 
points at a given place and time is the " magnetic north" for that 
place and time. The " magnetic meridian " of a place at a given 
time means the magnetic north and south direction at that time. 



COMPASS AND SEXTANT SUKVEYING 103 

Magnetic Declination. — As already defined, magnetic declination 
at a place at a given time is the horizontal angle which the magnetic 
needle makes with true north at that time, or it is the horizontal 
angle which the magnetic meridian makes with the geographical 
meridian. The declination at some places on the earth's surface 
lies to the west of true north, while at other places it lies to the 
east of true north. In Britain, in 1915, the declination was 
westerly throughout, varying from a maximum of about 22° 
on the west coast of Ireland to a minimum of under 15° at Yarmouth 
and Dover. Lines joining places of equal declination, known as 
isogonic lines, tend generally in a N.N.E. direction in Britain. 

For surveying purposes the direction indicated by the magnetic 
needle can be considered as constant only over a limited area. In 
passing from Yarmouth in a W.N.W. direction to the west coast of 
Ireland the direction of the needle changes on an average 1° for 
every sixty statute miles, or at the average rate of one minute per 
mile. 

The magnetic meridian at a given place is not constant, but varies 
continuously with time, and the magnetic declination alters cor- 
respondingly. The main variations of declination are (1) secular 
variations, (2) diurnal variations, (3) irregular variations. The 
secular variation is a gradual and continuous alteration of the 
magnetic declination which goes on from year to year with varying 
rate. The variation at London since 1580 is shown graphically 
on the following diagram (Fig. 112). The westerly declination 
of the needle is at present decreasing at all places in Britain. The 
British Admiralty publish a chart of the world showing lines of 
equal magnetic declination. 

The diurnal variation is a daily oscillation of the needle to the 
extent of a few minutes, usually less than six, on either side of its 
mean position. The daily oscillation is not constant in amount. 
It is greater in summer than in winter, greater in high latitudes 
than at places near the equator, and is variable from year to year. 

Irregular variations of large amount up to 1° or 2° sometimes 
occur suddenly, and are known as magnetic storms. 

Local Attraction. — The magnetic needle may be attracted and 
prevented from indicating the true magnetic meridian when it is 
in proximity to magnetite and certain eruptive rocks in the earth, 



104 



SURVEYING 



or to masses of steel or iron, such as bridges, rails, steel structures, 
water pipes, &c, or to electric cables or wires transmitting currents. 
Local attraction denotes any influence, such as the above, which 



YEAR 1320 
1900 
1880 
I860 
1840 
1820 
1800 
1780 
1760 
1740 
1720 
1700 
1680 
1660 
1640 
1620 
1600 
1580 



/ 



/ 



/ 






\ 



\ 



\ 



N 



30° 20' 10° 0° 10° 20° 30' 

West East 

Magnetic Declination at London. 

Fig. 112. — Diagram of Secular Variation at London. 

causes the needle to indicate something other than the magnetic 
meridian at a place. 

Bearing of a Line. — The bearing of a line is the angle less than 
90° which it makes with a north and south direction. The line 



COMPASS AND SEXTANT SURVEYING 



105 



AB (Fig. 113) makes two supplementary angles BAN and BAS 
with the north and south direction NS. The bearing of the line 
AB is the smaller angle BAN, and would be stated as 35° east of 
north, or N. 35° E. Similarly, the bearing of the line AC (Fig. 114) 
is the angle CAS or S. 40° E., the bearing of line AD is S. 45° W., 
and the bearing of line AF is N. 30° W. 

Magnetic Bearing is the bearing of a line with respect to the 
magnetic meridian at the place. 

True Bearing is the bearing of a line referred to the true north 
and south direction, that is, to the geographical meridian at the 
place. 

N 





Fig. 113. — Bearing of a line. 



W 



Fig. 114. — Bearings of lines. 



Whole Circle Bearing or Azimuth. — The whole circle bearing of a 
line is the clockwise angle between the north and south direction 
and the line, the angle being measured round from north as the 
starting point. Whole circle bearings run from 0° up to 360°. 
Referring back to Fig. 114, the whole circle bearing of the line AC 
is the angle NAC, or 140°. The whole circle bearing of AD is the 
angle NAD measured clockwise, and is equal to 180° + 45° or 
= 225°, and the whole circle bearing of AF is 330°. 

Forward Bearing and Back Bearing. — The bearing of a line, 
whether it be magnetic, true, or whole circle, differs according as 
the observation is made from the one end of the line or from the 
other. In Fig. 115 the bearing of the line AB taken from end A 
is seen to be N. 40° E., while taken from end B the bearing is 
S. 40° W. Note that the numerical value of the bearing is the 
same from both ends, but the designating letters are different. 



106 



SURVEYING 



In Fig. 116 the whole circle bearing of the line CD taken from C 
is 105°, while taken from D the whole circle bearing is 285°, the 
difference being 180°. Forward bearing and back bearing are 
terms used to distinguish between the bearings of a line taken 
from the different ends. If the bearings of a series of lines, such 
as AB, BC, CD (Fig. 117 a) are taken in order at their ends, pro- 
ceeding from A to D, then the bearing of each line taken at the 
end first arrived at will be considered a forward bearing, and the 
bearing taken from the opposite end will be a back bearing. The 
angles marked 1, 3, and 5 in Fig. 117 a represent the forward whole 




s s 

Fig. 115. — Forward 
and Back Bearing. 




Fig. 116. — Forward and 
Back Bearing (whole 
circle). 



circle bearings of the lines AB, BC, and CD, respectively, while 
the angles marked 2, 4, and 6 (Fig. 117 b) represent the back bearings 
of the same lines. 



Surveyor's Compass. — The surveyor's compass is employed to 
determine the direction of a line relative to the magnetic meridian, 
that is, to the direction in which the magnetic compass needle 
points. The utility of the instrument in surveying depends on 
the fact that within a limited area the direction of the magnetic 
meridian is almost constant. 

One of the forms, shown in plan in Fig. 118, consists of a circular 
compass box of brass or other non-magnetic metal, which is com- 
monly 6 to 9 in. in diameter, is covered with a glass lid, and contains 



COMPASS AND SEXTANT SURVEYING 



107 



the compass needle. A circle graduated to degrees or half degrees 
is engraved on a raised ring round the inner periphery of the box, 
the graduations being at the level of the top of the needle. The 
cardinal points E. and W. are marked on the circle in the reverse of 
their normal order, that is, in looking from S. towards N., W. is 
to the right hand and E. to the left hand. There are two zeros, 
one at the N. point and one at the S. point, and each quadrant is 
graduated from 0° up to 90°. A pivot with hardened steel point 
projects from the centre of the bottom of the case and supports 
the compass needle. The needle can be lifted of? its bearings 




Fig. 117. — Forward and Back Bearings of Lines. 

when not in use by means of a lever which passes through the side 
of the case. Hinged sight vanes, each having a vertical slot and 
a vertical hair sight, are attached to projections on the side of the 
case. The line of sight of the vanes is in the same plane as the 
north and south points of the card. In the better instruments 
one or two spirit levels are fixed to the box for levelling purposes. 
A vertical socket attached underneath to the centre of the case 
slips over a spindle fixed to the ball-and-socket joint at the apex 
of the tripod. The instrument can be rotated about the spindle 
and clamped in any position by a screw. The levelling of the instru- 
ment is done by turning it about the ball, which can also be 
clamped. 
In the instruments which are suitable for longer sights and finer 



108 



SURVEYING 



work the sight vanes will be replaced by a telescope, plate screws 
will be provided for exact levelling, as in the Dumpy level, and a 
vernier may also be provided for reading the angles. 

An inspection of the figure will show why the points are marked 
in the reverse order. It has to be remembered that the needle 
points in a constant direction and that the graduated circle rotates 
with the line of sight. When the line of sight, therefore, as shown 
in the figure, is directed towards the N.W., the needle indicates 




Fig. 118. — Surveyor's Comp 



N.W. on the circle as it should do. If the points were marked in 
their normal order, the needle would indicate N.E., which would 
manifestly tend to confusion. 

Prismatic Compass. — The prismatic compass is the most useful 
form of pocket or hand compass. In it the compass card is attached 
to the needle, and the north point of the card consequently always 
points to the magnetic north. The graduations, as shown in 
Fig. 119, run from zero at the S. point to 360° in a clockwise direction. 
There is a hinged vane at one side with hair sight, and at the 



COMPASS AND SEXTANT SURVEYING 



109 



opposite side of the case there is a vertical sight slot combined with a 
triangular glass prism which magnifies the graduations and enables 
the card to be read at the instant the sights are in correct line. 
The angle read is the whole angle of the line of sight, measured 
round from magnetic north in a clockwise direction. The reason 
for commencing the graduation of the card at the S. point in order 
to give the required angle will be evident from an inspection of 
Figs. 119 and 120. The line of sight is shown pointing N.W., the 




Fig. 119. 



-Graduation of Prismatic 
Compass. 



Fig. 120. — Explaining 
Graduation of Pris- 
matic Compass. 



whole angle of this direction measured round from magnetic north 
being 315°, and this is the card reading. 

The prismatic compass is useful for rough and rapid work. 

Compass on Theodolite. — This is shown in Fig. 121. The E. and 
W. points are marked in the reverse order, as in the surveyor's 
cornpass, and the graduations generally run from 0° to 360° in an 
anti-clockwise direction, starting from the N. point. The telescope 
is fixed in line with the N. and S. points. 

Methods of Graduating the Compass Card. — In addition to the 
methods of graduation above described other methods are some- 
times used. The surveyor's compass is sometimes graduated from 
0° to 360° in an anti-clockwise direction, starting from the S. point, 



110 



SURVEYING 



and sometimes it has both the complete circle and the quadrant 
graduation. Some of the older forms of compass have the points 
and graduations marked in the clockwise direction. Pocket com- 
passes (not the prismatic) are usually graduated from 0° to 360° in 
a clockwise direction, starting from the N. point, the graduated 
card being attached to and turning with the needle. In using 




Fig. 121.— Graduations of 



on Theodolite. 



any compass the method of graduation should be very carefully 
studied so that there may be no confusion when the angles read 
come to be made use of. 

Taking a Magnetic Bearing with the Compass. — To take the 
bearing of a line between two points, set up the compass over 
one of the points, level it, and clamp the ball. Let down the needle 
on to the pivot and turn the compass till the sights are brought 
into exact line with the distant point. The eye should, as far as 



COMPASS AND SEXTANT SURVEYING 111 

possible, be always applied to the same vane, preferably the one 
marked S., and the north end of the needle should be used as the 
index. Should the eye then on occasion be applied to the north 
vane the correct bearing will be obtained if the reading is taken at 
the south end of the needle. If the needle swings violently, its 
vibrations may be damped by raising the lifting lever so as just 
to touch it, and by tapping the case the effect of friction at the 
pivot may be diminished. In reading the graduation hold the eye 
vertically over the end of the needle, as otherwise an error of 1° or 
2° may readily occur in the reading. Bearings may be read to 
half or quarter degrees, according to the size of the circle. 

Advantages of the Compass. — The chief advantages of the compass 
arise from (a) the lightness and portability of the instrument and 
the rapidity with which it can be set up and a bearing taken ; (6) the 
fact that bearings are referred to magnetic north and the direction 
of each line is determined independently of that of any other line. 
The relative directions of lines within a limited area can be deter- 
mined, although the lines are not connected. There is no error 
carried forward from one line to another. 

Limitations of the Compass. — (a) With the ordinary compass 
bearings are not reliable to closer than about \° ; J° of error means 
a deviation of about 1 ft. in a length of 230 ft., so that the compass 
is only suitable for very rough work. 

(6) The needle is liable under certain conditions to give wrong 
indications. The bearing of a line cannot be reliably obtained in 
towns owing to the proximity of iron or steel and electric currents. 
Similarly, the needle is unreliable near railway lines, and may be 
affected by any small pieces of iron, such as knife blades, metal 
buttons, &c, carried by the observer, or by local causes within 
the earth. 

Use of Compass Surveying. — Compass surveying is only suitable 
on an extensive scale for rough work, where a considerable degree 
of accuracy is not expected or required. It, however, furnishes a 
very useful method of carrying out rapid prehminary work, such as 
for the approximate location of roads, railways, &c, in new 
countries, and is extensively used in mining work. 

It may also often be used to advantage in locating the details 



112 SURVEYING 

of a survey whose main lines have- been fixed by some more accurate 
method. Where numerous short bearings are required, the work 
will be much more rapidly overtaken with the compass than with 
the theodolite, and, provided care is taken to limit the extent of 
work surveyed independently by the compass, quite good results 
may be obtained. 

Compass Surveying with Needle only (Free Needle). — A system 
of survey lines is laid out in convenient proximity to the objects 
to be located. The arrangement of lines may take any shape, 
there being no limitation to a particular form, such as is necessary 
in chain surveying. A narrow strip of ground, such as a stretch 
of roadway, would be surveyed from a series of connected lines 
laid out in lengths to suit the windings of the road. In this case 
the accuracy of the work is mainly dependent on the accuracy 
with which the bearings and lengths are measured. There is 
nothing in the arrangement of the work to enable a mistake in 
measurement or bearing to be detected during plotting. 

A single enclosure would be surveyed from a set of lines 
forming a closed polygon. The lines may run around, within or 
along the boundaries as best suits the case. 

The lines to survey a considerable area should, where possible, 
be arranged in a series of closed polygons. 

Objects are referred to the survey lines by any of the methods 
described under chain surveying, and the compass itself may be 
used to furnish an additional method. By the latter method, 
which is very convenient for objects at some distance from the 
survey line, the compass bearing is taken to the object from a point 
on the survey line and the distance is also measured. Objects 
located in this way by bearing and distance are plotted by pro- 
tractor. As the general accuracy attainable in compass surveying 
is not great, refinement in locating details is unwarranted. 

The bearings of all the lines of a polygon may be obtained by 
setting up the instrument at each alternate angle. Thus, in the 
polygon ABCDEF (Fig. 122) all the bearings could be obtained by 
planting the compass at the points A, C, and E, the bearings of two 
sides being read from each point, and if the direction of the magnetic 
meridian could be relied on as being constant throughout, it would 
not be necessary to set up the compass at the other three points. 



COMPASS AND SEXTANT SURVEYING 



113 



There would, however, be nothing to show whether the indications 
of the needle were affected by local attraction at any point. 
By setting up the compass at all the angles the bearing of each 
line will be determined from both ends. If the bearing of a 
line taken at one end is not the exact converse of the bearing 
taken at the other end, the difference is probably due to local 
attraction. 

For methods of plotting the survey lines of a compass survey 
see Chapters XII. and XIII. 




Correction for Local Attraction. — Local attraction will affect 
equally the bearings of 
any two lines taken 
from the same point. 
Hence the correct angle 
between two lines can 
be obtained at their 
point of junction, even 
although their mag- 
netic bearings are 
individually incorrect. 
This points to a method 
of correcting the bear- 
ings in a system of 
lines where it is evi- 
dent that local attrac- FlG> 122 --Compass Bearings of a Polygon. 

tion occurs at only a few points. The table on p. 114 shows a 
set of bearings referring to Fig. 122. 

An inspection of the forward and reverse bearings of the various 
lines shows that there is evidently no local attraction at the points 
A, B, C, and F. The bearings of the lines C D and FE taken from 
the points C and F respectively may, therefore, be accepted as 
correct. The bearing of DC taken from D is 4° different from the 
converse bearing taken from C. The bearing of DC is, therefore, 
corrected to correspond with the bearing of CD. The bearing of 
DE must then be altered by the same amount as DC, and in such 
a way as to leave the whole angle CDE unaltered. The bearings 
of the other lines affected are dealt with successively, the corrections 
required being as shown in the table. 



114 



SURVEYING 



Table showing Correction of Magnetic Bearings 
Local Attraction. 



for 



Line. 


Observed Bearing. 


Correction. 


Corrected Bearings. 


AF 


N. 47° W. 




N. 47° W. 


AB 


S. 85° E. 


— 


S. 85° E. 


BA 


N. 85° W. 


, — 


N. 85° W. 


BC 


N. 15|° E. 


— 


N. 15*° E. 


CB 


S. 151° W. 


— 


S. 15|°W. 


CD 


S. 60° W. 


— 


S. 60 5 W. 


DC 


N. 64° E. 


-4° 


N. 60° E. 


DE 


N. 551° W. 


+ 4° 


N. 59|° W. 


ED 


S. 57° E. 


+ 2|° 


S. 59|° E. 


EF 


S. 271° W. 


-w 


S. 25° W. 


FE 


N. 25° E. 


— 


N. 25° E. 


FA 


S. 47° E. 


— 


S. 47° E. 



Booking the Survey. — It is most convenient to have the bearings 
recorded on a well-conditioned sketch of the survey lines. The 
stations are numbered or lettered and the details of each line are 
booked separately, as described under chain surveying. It is of 
much assistance in plotting to have the lengths of the survey line 
figured on the general sketch. 

As magnetic north is not a fixed but a varying direction, it is 
important that the magnetic declination or variation of magnetic 
north from true north should be accurately determined and recorded 
on the plan, together with the date of the survey. In newly- 
settled countries the preparation of survey plans and description 
of boundaries with reference to the magnetic north direction only 
has given rise to frequent disputes in later years owing to dubiety 
arising as to the true magnetic declination at the time and place of 
the survey. 



Surveyor's or Box Sextant. — This is a very useful hand instrument 
for measuring angles up to about 120°. It possesses the serious 
limitation for surveying purposes that it can only be used to give 
accurate results on ground which is level or nearly so. Fig. 123 
illustrates the principle of the instrument, and Figs. 124 and 125 
show a top view and horizontal section respectively. Two mirrors 



COMPASS AND SEXTANT SURVEYING 



115 



are employed, as in the optical square, to bring two separate objects 
into view simultaneously. A fixed mirror, A, and a movable 
mirror, B, are attached to the underside of the lid of the circular 
box frame of the instrument. The movable mirror, B, is connected 
to a spindle, which passes through the top of the case and carries 
the index arm which gives the angle on the graduated circle EF. 




Fig. 123.— Principle of Box Sextant. 

An eyehole at C in the side of the case is opposite a larger hole in the 
opposite side, and through these one of the objects, D, is seen 
directly over or under the mirror A. To bring the other object, 
say, G, into view in mirror A, mirror B must be rotated. Lines 
GBAC represent the path of a ray of light from object G when the 
image of the latter is seen in A. When mirror B is parallel to mirror 
A, as shown in dotted lines, the index arm points to the zero of the 
graduated circle, EF, and the reflected ray BD, from an object D, 

12 



116 



SURVEYING 



is parallel to the direct ray CD from the same object. The reflected 
image of D, therefore, appears in coincidence with the object seen 
directly. When the image of G appears in coincidence with D 
the angle included between the lines from the sextant to the two 
objects is GBD. As shown for the optical square, the angle GBD 
is double the angle EBF, through which the mirror B and the index 
arm turn. The graduated arc EF is, therefore, divided so that 
each \° is figured as 1°, or there are 180° to the quadrant instead 
of 90°. The whole instrument is contained in a case about 3 ins. 




Pig. 124.— Top View of Box Sextant. 

diameter. The index arm is furnished with a clamp and tangent 
screw for fine adjustment and has a vernier reading to single 
minutes. A small lens attached to a hinged arm enables the vernier 
to be read. Two darkened glasses can be interposed singly or 
together between the mirrors when readings are being taken to the 
sun. Provision is made for adjusting the fixed mirror A. It can 
be rotated a small amount in the horizontal direction by means of a 
square-headed screw in the side of the case, and a similar screw 
in the top of the case serves to adjust it vertically. A key to fit 
the heads of these screws is attached to the top of the case. A 



COMPASS AND SEXTANT SURVEYING 



117 



metal cover screwed on over the case serves to protect it from 
injury when not in use. 

Testing the Sextant. — (a) Testing for position of the image. To 
test whether the mirror A shows the image in a position favourable 
for accurate observation, set the index arm to zero, look directly 




Fig. 125. — Horizontal Section of Box Sextant. 

at an object and observe the position of its image in mirror A. 
If the image appears high above the object the instrument cannot 
be accurately used. Also, if when the image is visible the object 
itself is hidden by the mirror, the instrument is unusable. In either 
case, the mirror A requires to be adjusted vertically. To do this 
turn the square-headed screw in the top of the case until the image 
and object are seen simultaneously with the smallest possible 
vertical interval between them. 



118 SURVEYING 

(6) Testing for zero error. Set the index accurately to the zero 
graduation, hold the instrument level and sight directly to a distant 
definite object, preferably a vertical line, such as a ranging pole 
or the corner of a house. If the instrument is correct, the portion 
of the object seen in the mirror will appear in continuous vertical 
line with the portion seen directly. If the image appears slightly 
to one side of the object the instrument is in error at the zero point. 
To correct this apply the key to the screw in the side of the case and 
rotate mirror A until the image appears in exact vertical coincidence 
with the object. The sextant then indicates correctly at the zero 
point. 

(c) Testing for error at 90°. To test whether the instrument 
is correct at 90° set the index to 90 and use it as an optical square 
to set out two right angles from the same intermediate point on a 
straight base line, as explained on p. 19. If the two lines as set 

out coincide, the sextant is correct 

£ \ at 90°. If there is found to be 

// considerable error at 90° when the 

y/ zero is correct, the instrument is 

/ faulty and cannot be made right 

A B by the surveyor. 

Fig. 126.— Measuring large Angle The sextant may be tested 
with bextant. ., , . .. , 

throughout its range by comparing 

the readings of various angles with those given by a reliable theo- 
dolite. A faulty sextant may be used to give accurate results if its 
error is found at intervals throughout its range and the readings 
are corrected accordingly. 

Advantages of the Sextant. — The advantages of the sextant lie 
principally in its portability and accuracy, and the rapidity with 
which angles can be read. On level ground a 3-in. diameter sextant 
is much more accurate than any form of compass. 

Limitations of the Sextant. — (a) Smallness of the angle which can 
be read. The largest angle which can conveniently be read with 
the sextant is 120°. If the size of an angle greater than 120°, such 
as ABC (Fig. 126), is required, the line AB might be produced and 
marked at D, and the angle CBD read with the sextant. The 
required angle ABC would then be equal to 180° — CBD. Another 
method is to fix on some object E dividing the angle into two 






COMPASS AND SEXTANT SURVEYING 119 

portions, and read first the angle ABE and then the angle EBC. 
The sum of these angles gives the required angle ABC. 

(b) Accurate use is limited to ground which is nearly level. 
The sextant measures the actual angle between two lines in the 
plane containing the lines, while the angle required for surveying 
purposes is the horizontal projection of the actual angle. On 
sloping ground an angle measured with the sextant may be greater 
or less than the true horizontal angle. 



CHAPTER X 

THE THEODOLITE 

This chapter deals with the principles of construction of the 
theodolite, and with its use in the measuring of horizontal and 
vertical angles, ranging straight lines between survey stations, 
prolonging straight lines, and ranging survey lines between stations 
which are invisible from each other. 



Theodolite. — The theolodite is used by the surveyor for the purpose 
of accurately measuring horizontal, and vertical angles, and also 
for setting out angles on the ground, ranging straight lines, setting 
out curves, and for setting out the lines of intended works. It is 
the most important instrument used by the surveyor, and one with 
the principles of whose construction and use he must make himself 
thoroughly familiar if he is to become expert at his work. 

Before describing the theodolite in detail we shall indicate shortly 
its essential elements so as to give a general idea of the instrument 
as a whole (see Fig. 127). We may consider it as consisting of 
two main portions : the support or stand, usually in the form of a 
tripod, required to bring the instrument to a convenient height 
for the observer's eye ; and the upper or working portion. The 
upper portion is connected to the stand by three or four levelling 
screws, which permit of its being levelled up so as to make the axes 
of the rotating parts of the instrument truly vertical and horizontal. 
Immediately above the levelling screws there is a horizontal circle 
graduated to degrees and sub-divisions. This circle can rotate 
and can be fixed in any position by a clamp. Directly above the 
graduated circle and concentric with it there is a vernier or index 
circle which carries the standards supporting the telescope. The 
index circle, standards and telescope can rotate together hori- 
zontally on top of the graduated circle. The support of the tele- 
scope is by means of a horizontal axis resting on top of the standards 
so that the telescope can rotate independently in a vertical plane. 



THE THEODOLITE 



121 




Fig. 127.— Theodolite. 



122 SURVEYING 

For the reading of vertical angles there is a vertical graduated circle 
which is attached to the telescope and rotates with it. The vertical 
circle is read by means of a fixed index arm. 

A clamp and slow motion tangent screw are provided for con- 
trolling the motion of the whole upper working portion of the instru- 
ment, including the horizontal circle, and similar arrangements 
are provided for controlling the motion of the vernier circle and the 
parts above it relative to the horizontal graduated circle, and for 
controlling the motion of the telescope and vertical circle in the 
vertical plane. 

A pair of spirit levels at right angles to each other are provided 
on the vernier plate, or sometimes one on the vernier plate and one 
on a standard to show when the instrument is properly levelled. 
For use in reading vertical angles it is necessary to have a large 
spirit bubble attached either to the telescope or over the index 
arm of the vertical circle. 

The size of a theodolite is the diameter of its horizontal graduated 
circle in inches. Modern theodolites are mostly confined within 
the limits of the 4-in. and 10-in. size ; the most common size for 
ordinary use being the 5-in. reading to single minutes. 

In the form of theodolite known as the transit, which is the most 
common form, the standards are high enough to permit of the 
telescope making a complete revolution in the vertical plane, or of 
being " transited," as it is called. For the sake of compactness the 
standards are sometimes made so low that the telescope cannot 
turn right over or cannot be " transited," a restriction which 
greatly curtails the usefulness of the instrument for purposes of 
ranging out fines and setting out works. In the non-transiting 
form of theodolite (plain theodolite) the complete vertical circle 
is replaced by either a single graduated arc under the telescope 
with a single vernier, or by two graduated arcs of a circle diametri- 
cally opposite each other with a vernier to each. 

Support or Stand. — The most common form of support or stand 
for the theodolite consists of a tripod formed of three tapering solid 
wooden legs, pointed and shod with iron at the lower ends and hinged 
at the top to a brass or gun-metal casting, which is screwed to 
receive the upper portion of the instrument. The legs are of 
rounded triangular section, so that they take up a circular form 



THE THEODOLITE 123 

when closed. They are bound together by brass rings or leather 
straps when not in use, and a cap is provided to screw on to the head 
of the tripod and protect the threads from injury. 

In situations where the tripod cannot be set up a support of the 
form shown in Fig. 128 is sometimes useful. It may be set on the 
level top of a wall or a level board may be readily fixed up to carry 
it in places where much trouble would be required to construct a 
stand for the tripod. 

The theodolite illustrated in Fig. 127 is supported by the framed 
type of stand, which, when properly and substantially constructed, 
is preferable to the solid wooden type in respect of stiffness. In 
this type each leg of the tripod consists of two members, which 
incline downwards towards each other, and are joined together at 
the bottom and shod to form a single point of support. The two 
members forming a leg are braced 
together at one or more intermediate 
points. The framed type of stand 
is almost universally used for large 
theodolites. 




Parallel Plates. — The lower parallel 
plate either screws on to or forms 
part of the head of the tripod, and Pm 128 ._ Triangular stand . 
is a fixed portion of the instrument. 

In the four-screw form of construction the connection between the 
upper and lower plates is through a ball-and-socket joint and the 
thumb -screws. These are screwed into the upper plate and simply 
rest on the lower plate, the foot of one of them fitting into a cup 
to prevent rotation of the upper plate. 

The upper parallel plate contains the socket which receives the 
spindle supporting the whole rotating portions of the instrument. 
The thumb -screws provide the means of levelling the upper portion 
of the instrument and rendering its vertical axis truly vertical. 
In modern instruments the parallel plates are generally castings 
with three or four arms for the fixing of the levelling screws, and 
have little resemblance to plates. 

Three-Screw Levelling Arrangement. — In the levelling arrange- 
ment illustrated in Fig. 127 there are three thumb -screws at 120° 



124 SURVEYING 

apart. In this case the screws are the only connection between the 
upper and lower plates, and themselves serve the function of a ball- 
and-socket joint. At the foot of each screw there is an enlarged 
ball or cone which engages in and is held by a groove or recess in 
the lower plate in such a manner as to permit of the screw tilting 
to the required extent. The upper ends of the screws are threaded 
and engage with the arms of the upper plate. The three-screw 
levelling arrangement is preferable to that with four screws. In 
the four-screw type, racking and straining of the screws will readily 
occur unless care is taken to turn each of a diagonally opposite 
pair at uniform rates in opposite directions. A single screw cannot 
be separately screwed down without causing strain. In the three- 
screw type any single screw can be turned (within reasonable 
limits, of course) without straining the instrument, and the levelling 
can be completed by turning the screws one at a time with one hand 
if necessary. 

Graduated Horizontal Circle. — The graduated horizontal circle is 
attached to the spindle which rotates within the socket of the upper 
parallel plate. The graduations are engraved on a bevelled ring 
of silver laid round the rim of the circle. In theodolites for ordinary 
purposes, such as that illustrated in Fig. 127, the circle is generally 
5 ins. in diameter. The graduations are |° apart and run from 
0° to 360° right round the circle in a clockwise direction. Attached 
to the upper plate there is a clamp with locking key which engages 
with a portion of the spindle connected to the graduated circle. 
The attachment of the clamp to the upper plate is through a tangent 
screw, so that when clamped the circle can still be turned a small 
amount under the exact control of the observer. 

Vernier Plate. — The vernier or index plate is concentric with, 
and rotates on top of the graduated circle. It usually has two 
indexes at the opposite ends of a diameter with verniers, enabling 
the circle to be read to single minutes, and is frequently arranged 
to cover and protect the graduated circle except for a small portion 
opposite each index. The vernier plate can be clamped to the 
graduated circle, and the connection is made through a slow- 
motion tangent screw which enables the index to be set accurately 
to any desired angle. Magnifying glasses are provided for reading 
the verniers. 



THE THEODOLITE 125 

The vernier plate carries on its upper surface the two standards 
of A-frame or other form which support the horizontal axis of the 
telescope, and two spirit levels, one placed parallel to the hori- 
zontal axis of the telescope and the other at right angles thereto. 
The latter spirit level is sometimes carried on one of the standards 
instead of on the upper plate. The spirit levels are attached to the 
plate or standard by capstan screws, cc, which enable them to be 
adjusted so as to indicate correctly. 

A magnetic compass with a small circle graduated as described 
on p. 109 is usually also carried on the upper plate between the 
standards. Instead of the compass on the upper plate a trough 
compass with a long and sensitive needle is sometimes fixed to the 
underside of the graduated circle of the theodolite. The trough 
is a narrow box with a very short portion of graduated arc at each 
end. The line of sight of the telescope is parallel to the line joining 
the zeros of these arcs, so that the direction of the magnetic meridian 
is obtained when the needle indicates zero. 

Standards. — The horizontal axis of the telescope rests in V-shaped 
bearings on the top of the standards and is held down by hinged 
clips fastened by thumb -screws. One of the bearings can be ad- 
justed vertically by means of the screws a, a. 

Arrangement of Telescope, Vertical Circle, &c. — The telescope is 
fixed at right angles to its horizontal axis, and the vertical circle 
is rigidly attached to the telescope and rotates with it. The verniers 
are at the end of a horizontal arm which is formed in one piece 
with a vertical clipping arm, the latter being attached to a pro- 
jection on one of the standards by means of opposing screws. The 
horizontal axis of the telescope passes through the junction of the 
clipping arm with the vernier arms. A clamp and tangent screw 
attached to the vertical arm serve to fix the vertical circle when 
required and give it a slow rotation for accurate setting of the 
telescope. The complete vertical circle attached to the telescope 
of a transit theodolite has two zeros, which can be made to coincide 
with the indexes of the verniers when the fine of sight is horizontal, 
each quadrant being graduated from 0° up to 90°. Sometimes 
there are four zeros at the extremities of two diameters at right 
angles to each other, and each quadrant is then divided up from 
0° to 90° in a clockwise direction. The vertical circle is also 



126 



SURVEYING 



sometimes graduated from 0° to 360° in 
a clockwise direction, giving 0° at one 
vernier and 180° at the other when the 
telescope is level. With this method of 
graduation angles of elevation can be 
distinguished from angles of depression 
without remark if the telescope is not 
transited. 

A sensitive spirit level is attached to 
the upper or under side of the telescope, 
or, as shown in Fig. 127, to the vernier 
arm. The line of sight is arranged 
parallel to the axis of the bubble tube 
(see Chap. XVI.), so that it will be hori- 
zontal when the bubble is at the centre 
of its run. The attachment of the spirit 
level to the telescope is by means of 
opposing capstan screws which permit of 
its accurate adjustment. 



Telescope. — The essential parts of the 
telescope are the tubes, the object glass, 
the magnifying eyepiece, and the dia- 
phragm with hair sights. The arrange- 
ment of the telescope is shown in Fig. 129. 
The object glass is fixed in the end of 
a tube which slides within the main 
telescope tube and has a range of motion 
of 1 or 2 ins. The eyepiece also slides 
within a contracted portion of the main 
tube, with a very much smaller range 
of motion than the object glass. Fre- 
quently, however, the object glass end 
is fixed and the motion for focussing 
takes place at the eyepiece end. The 
diaphragm is fixed in the main tube near 
the eyepiece end. The object glass or 
eye end, as the case may be, is moved 
in and out by turning a thumb screw on 



THE THEODOLITE 127 

the side of the telescope, which works a rack and pinion inside 
the tube. The object glass is a combination of two lenses placed 
in contact. The front lens is double convex and made of crown 
glass. The back lens is of flint glass and plano-concave, that is, one 
side is plane and the other is hollowed to fit against the front lens. 
The combination has very nearly the same effect as a single double- 
convex lens, but gets rid of certain optical imperfections which are 
unavoidable with the single lens. Rays of light coming from a 
point in front of the object glass and passing through it converge 
to a point in the vicinity of the diaphragm. Rays of light from an 
object in front of the object glass converge on passing through it and 
form a small inverted image of the object near the diaphragm. The 
focussing of the object glass consists in moving it or the eyepiece 
end in or out until this image coincides with the cross hairs. 

The eyepiece generally consists of two small plano-convex lenses, 
placed as shown in the figure, the combination forming a microscope 
with which the cross hairs and coincident image of the object are 
viewed. With such an eyepiece the object appears inverted, that 
is, it appears upside down and the right-hand side appears to the left 
hand. To focus the eyepiece on the cross hairs, sight the telescope 
towards the sky and move the eyepiece in or out until the cross 
hairs appear perfectly distinct. It should then be in correct focus, 
and should not require to be again altered for any sight taken by 
the same observer. When the eyepiece has been correctly focussed 
on the cross hairs, sights may be taken with the telescope. To 
bring an object into view, start with the object glass in its furthest 
in position and turn the thumb screw to move it slowly outwards. 
A point will be reached at which the image appears brightest and 
most distinct, and the object glass is then correctly focussed. 
The cross hairs and image should now both appear clear and definite, 
and if the eye is moved from side to side the cross hairs should not 
appear to move relative to the object. If there is any apparent 
motion the focussing is imperfect, but very slight motion of the 
object glass should now suffice to give the desired result. Apparent 
motion, caused by the image and cross hairs not being in the same 
plane or by the cross hairs not being in the focus of the eyepiece, 
is known as parallax. Accurate work cannot be done unless the 
parallax is got rid of, as otherwise the angle read would depend on 
the position of the eye. 



128 SURVEYING 

The diaphragm most commonly consists of a brass ring within 
the telescope tube, across the aperture of which three spiders' webs 
or very fine platinum wires are stretched. These are known as the 
cross hairs. One is horizontal and the other two are equally inclined 
to the vertical, one on each side of it, as shown in Fig. 130. They 
cross each other at the centre of the aperture. Sometimes fine 
lines scratched on glass or platinum-iridium points are used as a 
substitute for the hairs. The diaphragm ring is fixed inside the 
tube by two or four screws, whose capstan heads appear outside 
the tube. These screws pass loosely through enlarged holes in the 
tube and are screwed into the diaphragm ring, affording the means 
of adjusting the latter slightly in the vertical or horizontal directions. 
The " line of sight " of the telescope is the fine joining the inter- 
section of the cross hairs to the optical 
centre of the object glass. The motion of 
the centre of the object glass in focussing 
should be along that line, which should 
pass as nearly as possible through the 
optical centre of the eyepiece. The term 
" line of collimation " is often used with 
the same meaning as " line of sight." 

Vernier. — The vernier, so called from 
Fl °'aid°G^s?H&™ 8m the > me of its inventor, is a device which 
enables a scale to be read accurately to a 
small fraction of its smallest division. Its practical utility is largely 
due to the extreme accuracy with which scales can be machine 
divided, and depends also on the fact that the eye can judge very 
accurately when a line on one scale is exactly opposite a line on 
another when their graduated edges are in coincidence. The 
vernier consists of a short graduated scale whose edge slides against 
the edge of the scale to be read. The length of the vernier scale 
depends on the fineness with which it is desired to read the main or 
fixed scale. In order that the main scale should be read to the 
tenth part of its smallest division, the vernier scale is made exactly 
equal to nine of the main scale spaces and is divided into ten equal 
parts. To read to the thirtieth part of the main scale divisions 
the vernier scale would have thirty equal divisions in a length 
equal to twenty-nine main scale spaces. 




THE THEODOLITE 



129 



Fig. 131 illustrates a vernier to read to the tenth part of the 
scale divisions. As shown in (1) the vernier has a length equal to 
nine scale spaces, is divided into ten equal parts, has an index at 
one end, and has its graduations numbered, the zero being at the 
index. The vernier is indicating 5-00 on the scale. Each main 
scale space has a length of 0-10 unit. Each division of the vernier 

scale has a length equal to 



0) 



10 



or 0-09 unit. The difference in 



Hill 



Nihil 



1-1 



HE 



I I I I 1 M 



(?) \ i i i i 



h 1 l I. Uu xl 



H ! i i -J ' ' i | i | 



ii i i 



T U . I . I J J ITTT 



Mill 



(3) 



TIT I 



V 



/»;jT I I I | I I I I | 



?ffft 



1 



II I I 



■I' l l 



i 



1 1 1 rriT 



XXI 



i 



(5)\ 



I I I I | 



Fig. 131.— Vernier Reading to Tenths. 

length between a main scale division and a vernier scale division 
is, therefore, = 0-10 — 0-09, or = 0-01 unit. When the vernier 
index then coincides with the line marked 5 on the scale, the 
next line on the vernier will be 0-01 short of the next line 
on the scale, the second vernier line will be 0-02 short of 
the second scale line past the number 5, and so on. If, there- 
fore, the index is moved forward 0"01 unit, or one-tenth of a 
scale division, the vernier line, No. 1, will come opposite a scale 

S. K 



130 SURVEYING 

division, and the scale reading is then 5-01. If, as shown in 
Fig. 131 (2), the index has been moved forward two-tenths of a 
division from the line marked 5, the vernier line No. 2 will coincide 
with a scale division, and the reading will be 5-02. Fig. 131 (3) 
shows vernier line No. 7 in coincidence with a scale line, and the 
reading is 5-07. The index in Fig. 131 (4) records 5-2 plus a fraction 
on the main scale. Vernier mark No. 6 is opposite a scale mark. 
The scale reading is, therefore, 5'26. 

Any other vernier is read in a precisely similar manner. 

It may happen that no one line on the vernier is exactly or nearly 
opposite a scale line, but that two adjacent vernier lines appear 
equally close to two lines on the scale. This is illustrated in 
Fig. 131 (5), where the two vernier lines, Nos. 4 and 5, are sym- 
metrical with respect to two of the scale lines. In such a case the 
average is taken, the reading indicated being 5*445. 

Theodolite Verniers. — In the graduated circle of the ordinary 
5-in. theodolite the lines are engraved at %° intervals, so that each 
space represents thirty minutes. The vernier reads to single 
minutes, or one-thirtieth of the smallest scale division, and therefore 
its length is made equal to twenty-nine circle spaces, and this length 
is divided into thirty equal parts. 

A theodolite vernier reading to minutes is illustrated in Fig. 132. 
In Fig. 132 (1) the length of the vernier scale is seen to be equal to 
14|° or twenty-nine spaces of the circle. The reading is 20° 0'. 
Fig. 132 (2) shows the index pointing between 22° and 22|°, and the 
vernier mark, No. 12, is opposite a scale graduation. The reading 
indicated is, therefore, 22° 12'. Fig. 132 (3) shows the index pointing 
between 25|° and 26°, and vernier mark No. 27 coincides with a scale 
mark. The reading is, therefore, 25° 30' + 27' or 25° 57'. A 
common mistake when one is intent on reading the vernier cor- 
rectly is to neglect to add the |° in angles such as the latter, the 
result being that 25° 27' is booked instead of 25° 57'. 

In theodolites with circles larger than 5 ins. diameter the gradua- 
tions are usually at intervals of ^°, and the vernier reads to one- 
third of a minute or twenty seconds. The vernier, therefore, reads 
to the one-sixtieth part of the smallest circle space and requires 
to have sixty divisions in a length equal to fifty-nine circle spaces. 
Every third mark on the vernier scale represents a whole minute 



THE THEODOLITE 



131 



and is drawn longer than the others. These lines representing 
minutes are numbered from up to 20, and the intermediate short 
lines represent twenty seconds or forty seconds. Fig. 133 shows 






o 




CD 


K> 




3 

1 

o 




fe 


60 

.a 

•X3 




fc"l 






l\l 






o> 


Ph 




a vernier reading to twenty seconds, drawn much larger than actual 
size. In Fig. 133(1) the reading is 30° 0', and it is seen that the vernier 
scale covers 19° 40' of the circle. In Fig. 133 (2) the reading is 
30° 12'. In Fig. 133 (3) the index is one space and a fraction past 

k2 



132 



SURVEYING 



the whole degree mark and twenty minutes must, therefore, be added 
to the vernier reading. The reading shown is 32° 20' + 7' 40", 
or 32° 27' 40". In Fig. 133 (4) the index is two spaces and a 











o 


-^- 


= 




tn 










= 






— 


















zz. 












— 








— 








= 




















o 




;ZZ 


o 








^t 










































































~ 






— 


ZZ 













o .* 




fraction past the whole degree mark and forty minutes have to be 
added to the vernier reading. The reading shown is 35° 40' 
+ 13' 20" or 35° 53' 20". With this vernier care must be 
taken not to drop 20' or 40' from the correct reading. 



THE THEODOLITE 



133 



<?> 




I'M 


CM 




o 


Ui 


to 


ro 




nn! 




Pn 


G 





In the vernier scales shown in Figs. 132 and 133 there is an extra 
mark at each end beyond the scale proper. These extra marks 
facilitate the reading when the index 
is nearly opposite a scale division, and 
are specially useful in setting the index 
to zero or to a given angle. The 
index can be most accurately set by 
looking at the marks on each side of it, 
and noting when these are exactly sym- 
metrical with respect to the adjacent scale 
marks. 

Before reading any vernier it is a good 
plan to estimate directly from the posi- 
tion of the index on the scale the approxi- 
mate number of minutes. This can be 
done to within a few minutes, and forms 
a rough check, which tends to the avoid- 
ance of large mistakes. At the same time 
it enables the vernier to be read more 
rapidly, since the approximate position 
of the mark is known, and it need not be 
looked for elsewhere. 

There are two verniers diametrically 
opposite each other on most theodolites. 
Only for work of great precision are both 
verniers made use of, and then in the 
manner explained in Chapter XIV. In 
ordinary work only one vernier is read, 
and it is important that the same vernier 
should be used throughout, as the verniers 
are not always exactly 180° apart. The 
two verniers are generally distinguished 
by the letters A and B. Always use 
the A vernier. Use the B vernier only when both are being read. 

Use of the Theodolite. 

Use of the Theodolite. — In using the theodolite the preliminary 
operations consist of setting up the instrument over a given point, 
centering it exactly over the point by means of the plumb-bob, 



o — 

x- ~ 

ol= 

<M ±Z- 


— O 

Ki 



* B 



134 SURVEYING 

and thereafter levelling it by means of the plate screws and spirit 
levels. 

Setting up the Theodolite. — The parts of the head of the theodolite 
are carried in a box which is accurately fitted and padded to prevent 
any looseness or motion of the parts. There is usually only one 
definite position and state of adjustment in which the parts will 
go into their allotted places. It is therefore very important, 
before taking the portions of a theodolite out of the box for the 
first time, to make a careful note of their position and arrangement 
in the box. It will be found very useful to make a sketch of the 
arrangement on the inside of the lid of the box. The 5-in. theodo- 
lite, without sliding head, is usually packed into the box in two 
main portions. The lower portion which screws directly on to the 
tripod head comprises the parallel plates, the horizontal graduated 
circle, the vernier plate, and the standards. The upper portion 
consists of the telescope vertical circle, &c. 

Having set up the tripod and unscrewed the protecting cap, 
lift the lower portion of the instrument carefully out of the box and 
screw it on to the tripod head. Bring all the plate screws to a 
bearing on the lower plate, see that they are equally screwed up, 
and then open the clips over the telescope bearings at top of the 
standards. Take now the telescope portion, bring the vertical 
clamping arm between the standards and on to its attachment, 
at the same time lowering the horizontal axis on to its bearings. 
Bring the opposing screws of the vertical clipping arm attachment 
to a hold and fasten the clips over the ends of the horizontal axis. 

The theodolite is carried from place to place on the shoulder. 
To avoid straining the instrument from the unavoidable jolting in 
carrying it, the lower clamp should be loose, so that the whole head 
may be at liberty to rotate. The vernier plate clamp may also be 
loose. The telescope should be pointed vertically upwards and 
clamped. In passing under trees with low branches or through 
low passages the theodolite should be carried under the arm with the 
head in front. 

Setting over a Point. — See that the upper and lower plates are 
nearly parallel, and, if necessary, turn the thumb-screws to effect 
this. See also that the connections of the legs to the tripod head 
are sufficiently stiff. These connections in some makes of instru- 



THE THEODOLITE 



135 



ment are apt to become loose when the theodolite is in constant 
use, and accurate work is then impossible. A screw-driver or key 
should be kept handy for tightening the screws. Sometimes the 
cap of the tripod is formed as a key to fit the nuts. 

The plumb-bob is suspended from a hook in the central axis 
of the instrument and a sliding knot or other arrangement, such 
as illustrated in Fig. 134, is employed in adjusting its height. The 
knots are shown open for the sake of 
clearness in illustration, but in practice 
they are pulled tight enough to prevent 
the plumb-bob from falling under its 
own weight. The arrangement shown in 
(b), Fig. 134, is perhaps the simplest. 
Both the loop and the free end of the 
string are pulled down vertically so that 
the portion carrying the plumb-bob is 
gripped with sufficient tightness to pre- 
vent it from slipping. In the sliding 
button arrangement in (c) an actual 
button may be used, or a flat piece of 
metal, wood, or leather with holes for the 
attachment and passage of the string will 
equally serve the purpose. 

To bring the instrument nearly into 
position over a station it will be found 
useful to grasp the forward legs, one in 
each hand, and allow the back leg to 
pass the right thigh with its shoe trailing 
on the ground, and, thus held, move the 
instrument bodily about, keeping the eye 
on the plumb-bob and on the plate levels, 
points of the tripod into the ground, move them slightly in or 
out or laterally as required to bring the plumb-bob almost over 
the exact point. In doing this it has to be remembered that 
a motion of a tripod shoe outwards or inwards in the direction of the 
leg causes a corresponding motion of the plumb-bob (but only of 
about half the amount) in the same direction, and on fairly level 
ground does not much alter the inclination of the parallel plates, 
while the swinging of a shoe in an arc of a circle about the plumb-bob 




134. — Suspension of 
Plumb-bob. 

Then, before pressing the 



136 SURVEYING 

as centre alters the inclination of the parallel plates, but hardly 
shifts the plumb-bob. Only with practice can the surveyor become 
adept at swinging the legs and moving them simultaneously in or 
out, so as to arrive at accurate centering, keeping the parallel 
plates at the same time nearly level, in the shortest possible time. 
The final adjustment of the plumb-bob over the station when the 
theodolite is set up on the ordinary earth surface is effected in 
pressing the shoes into the ground. The shoes should be pressed 
in by walking round the theodolite and taking the legs one after 
the other with both hands and applying a force directly along the 
leg towards its point. The application of any force tending to 
bend the legs should be avoided, and the surveyor should never 
attempt to push in a leg while standing at the other side of 
the instrument. 

On steeply sloping ground the setting up will be most readily 
and quickly accomplished if two legs are placed downhill and one 
uphill. If this precaution is not observed it will be found very 
difficult to get the head on the instrument level. 

On pavements and hard, smooth surfaces precautions require 
to be taken to prevent the shoes from slipping. They should be 
placed, where possible, in cracks or joints, and it may sometimes 
be necessary to make a nick for the shoe with the point of a pole 
or with hammer and chisel. Looseness of the joints at the tripod 
head produces the worst effects when the theodolite is set up on a 
hard surface. 

It is sometimes necessary, especially in the setting out of works, 
to measure with chain or tape to a station while the theodolite is 
set up over it. It is then important to see that the instrument is so 
placed that none of the legs obstruct the chain fine. 

If the theodolite is provided with a shifting head the final adjust- 
ment over the station is much more rapidly effected. Get the 
plumb-bob to within about f in. of the exact point, with the legs 
firmly planted and the parallel plates sufficiently level, and then 
move the plumb-bob the rest of the distance by means of the 
shifting head. 

The more carefully the theodolite is set up in the first instance 
the less time will be expended in levelling up by the thumb-screws 
and the less will be the wear and tear of the instrument. The 
knack of rapid and accurate setting up should be sedulously 



THE THEODOLITE 137 

cultivated by the aspiring surveyor, and can only be acquired 
by practice. 

Levelling Up. — To level up the theodolite start with the lower 
clamp or with both clamps loose. Turn the upper portion of the 
instrument so as to bring each vernier plate level parallel to a pair 
of diagonally opposite levelling screws where there are four screws. 
Bring the bubble of one of the levels to the centre of its run by 
turning the pair of screws parallel to it at a uniform rate in opposite 
directions. See that both screws remain bearing on the lower plate 
without becoming tight. Bring the bubble of the second level to 
the centre of its run in the same way by turning the second pair of 
screws. This will probably disturb the first level somewhat, and 
the levelling of each must be repeated till both bubbles are central ; 
they should then remain central when the instrument is rotated. 
If, as is generally the case, one of the levels is more sensitive than 
the other, it should be used exclusively for the final levelling. Turn 
it first parallel to one diagonal pair of screws and then parallel to 
the other, levelling it each time until it remains level when the instru- 
ment is rotated. 

When there are only three levelling screws, turn one of the levels 
parallel to a pair of screws and bring its bubble to the centre of its 
run by turning these screws. Bring the bubble of the other level 
to the centre of its run by turning the remaining screw and repeat 
till both bubbles are central. 

It is not advisable to spend time in bringing the bubble of the 
first level exactly to the centre of its run. Get both bubbles rapidly 
to an approximately central position and then proceed to level 
them exactly. 

The levelling screws must be handled very carefully. If a 
screw begins to work tight it should never be forced. In the four- 
screw arrangement the stiffness of one pair of screws is sometimes 
relieved by slacking back both of the other pair of screws a little. 
When the tightness is due to the instrument being set up very 
much off the level, the only remedy is to plant afresh. 

Measuring Horizontal Angles. — It is required to read the hori- 
zontal angle ABC (Fig. 135) with the theodolite, which has been set 
up and levelled at point B. The upper and lower clamps are loose. 
Turn the vernier plate and graduated circle relative to each other 



138 SURVEYING 

till the index of vernier A is nearly opposite the zero of the circle. 
Set the upper clamp, thus fixing the vernier plate and graduated 
circle together, then turn the upper tangent screw till the index 
coincides exactly with the zero. For the final setting, to avoid 
parallax, make sure that the centre of the reading glass is vertically 
over the index, and see that the lines to each side of the index are 
symmetrical with respect to the circle divisions. 

Now apply the fingers of both hands to the edge of the graduated 
circle and turn the head of the instrument till the telescope is 
pointing nearly to A, as judged by sighting with the eye along the 
top of the telescope. Then look through the telescope, focus it 
on the sighting mark at A, and adjust it by hand so that the cross 
hairs are as nearly as possible on the mark in both the vertical and 
horizontal direction. Then fix the lower clamp and bring the cross 
hairs horizontally into coincidence with the mark by turning the 
lower tangent screw. If necessary, 
fix the vertical circle clamp and use 
its tangent screw for the vertical 
adjustment. The telescope is now 
pointing to object A, the upper and 
B c lower clamps are fixed, and the vernier 

Fig. 135.— Measuring a Hori- indicates zero. To read the angle 
zontal Angle. between the lines BA and BC, the 

lower clamp therefore remains set, the upper clamp is loosened and 
the telescope is turned in a clockwise direction and pointed towards 
by applying the fingers of both hands to the edge of the vernier 
plate or the foot of the standards. Set the centre of the cross hairs 
exactly on point C by fixing the vernier plate clamp and using its 
tangent screw. The angle ABC can now be read off from vernier A. 
It is not essential that the vernier should first be set to zero in 
order to read a horizontal angle. This is done for convenience 
and in order to get a direct reading of the angle. The telescope 
may be pointed to A with the vernier set at random. Read off 
the angle indicated by the vernier and with lower clamp fixed, 
upper clamp loose, turn the telescope on to C and again read the 
angle. The difference of these angles gives the angle ABC, but if 
the vernier in passing from the first to the second position has 
crossed the zero of the circle, 360° must be added to the second 
angle before the subtraction is made. 



THE THEODOLITE 139 

In reading a series of adjacent angles from the same point the 
vernier would, in general, be set to zero for the first sight. Then, 
without shifting the lower clamp, the telescope would be turned 
in succession towards the other marks and the angles read off from 
the vernier. The angles so read would be angles measured round 
in a clockwise direction from the line of the first sight. The actual 
angle between any two adjacent directions would be got by sub- 
traction. 

If the bearing of line BA is known with respect to some reference 
direction, such as magnetic or true north, and it is required to find 
the bearing of line 1$C with respect to the same direction, the 
telescope should be pointed to A with the vernier set to indicate 
the bearing of BA. The angle read off when the telescope is pointed 
to C will then be the bearing of BC. 

Measuring a Vertical Angle. — (a) Angle of elevation or depression 
from a horizontal plane. The theodolite having been planted 
and levelled up, fix the vernier plate clamp and leave the lower 
clamp loose. Bring the telescope nearly to the level, fix the upper 
circle clamp, and by turning the tangent screw bring the zero of the 
upper circle to exact coincidence with the vernier index. The 
telescope bubble should be nearly level. Bring it exactly to the 
level by turning the opposing screws which hold the clipping arm 
to the attachment on the standard. This will cause the rotation 
together of the index arms, vertical circle, and telescope, so that 
when the telescope is level the indexes are at zero and the instru- 
ment is in readiness for reading an angle of elevation or depression. 
Then, to read such an angle, loosen the vertical circle clamp, sight 
the telescope on the object, clamp the vertical circle and adjust the 
horizontal cross hair exactly on to the point by turning the tangent 
screw. The required angle is then given on the vertical circle. 

The adjustment to make the telescope bubble remain exactly 
central as the head of the instrument is rotated is described in 
Chapter XXII. 

Instead of adjusting the vernier to read zero when the telescope 
is level its actual reading may be noted and treated as an index 
error. Any angle read off the vertical circle would then require to 
be corrected by adding or subtracting the amount of this error 
according as it occurred on the opposite or the same side of the zero. 



140 SURVEYING 

(b) Vertical angle between two points. To measure the vertical 
angle between two points sight the telescope on one of the points, 
bring the horizontal cross hair exactly to the mark, and read the 
angle on the vertical circle. Then sight to the second point and 
again read the angle. The difference or sum of these angles will 
give the required vertical angle according as they occur on the same 
or on opposite sides of the zero. 

Measuring Angles by Repetition. — By repeating the measurement 
of an angle several times in such a way as to add up the successive 
readings on the graduated circle a more accurate determination of 
the angle can be made than by a single reading. The method is 
described in Chapter XIV. 

Doubling the angle forms a useful check to ensure that no serious 
mistake has been made in the reading. The angle is measured 
once and its value booked. The angle is then repeated and the 
circle reading is noted and divided by two. This should give a 
value almost the same as the first reading, and any serious error 
would be at once apparent. 

Ranging a Straight Line. — With the aid of a theodolite, survey 
lines can be ranged out with almost perfect accuracy, and straight 
lines can be set out across hills and hollows with little trouble. To 
set out points in line between two survey stations for the guidance 
of the chainmen the theodolite will be set up and levelled over one of 
the stations. Particular attention should be paid to the vertical 
adjustment of the instrument, and to the level which is parallel 
to the horizontal axis of the telescope. With the vernier plate 
clamp fixed and the lower clamp loose, sight the telescope on to 
the bottom of a pole held at the distant station. Fix the lower 
clamp and use the lower tangent screw to bring the centre of the 
cross hairs exactly on to the pole. Both upper and lower clamps 
are now fixed and must remain so while the points are being lined 
in. The telescope can turn vertically on its horizontal axis so that 
its line of sight moves in a vertical plane passing through the 
two stations, and any point where the line of sight strikes the 
ground will lie on a straight line between the stations. Poles 
should be planted in order coming from the distant station towards 
the theodolite, so that when set they may not obstruct the view 
of the next point. In directing the assistant into line the observer 



THE THEODOLITE 141 

at the theodolite looks first along the top of the telescope and 
signals the assistant to move in the required direction. When he 
comes into the field of view the observer looks through the telescope, 
gets the pole into focus, and, sighting as near the ground as possible, 
directs the assistant to move the pole laterally till it is bisected by 
the centre of the cross hairs. He then gives the signal to plant. 

In ordinary surveying work there is no need for great refinement 
in setting the poles in fine. The thickness of a pole will hardly 
affect the accuracy of the survey. In town work, however, and in 
the setting out of works, accurate alignment is essential, and the 
theodolite should be sighted on a fine mark, such as is afforded by 
the point of a pencil or arrow or the string of a plumb-bob. 

To fix an intermediate survey station in line between two main 
stations the peg to be driven should, where practicable, be sighted 
to instead of a pole. The observer at the theodolite should watch 
the peg as it goes down and direct the driving so that the middle of 



Fig. 136. — Prolonging a Line. 

the peg is kept nearly in line. If accurate lining is necessary, or if 
the distance between pegs is short, a small nail should be lined in 
on the head of the peg and driven to mark the exact point. When 
the position of the peg is invisible from the theodolite owing to 
long grass or otherwise a ranging pole or the string of a plumb-bob 
should be fined in and the position of the station marked on the 
ground. The peg may then be driven and the plumb-bob used to 
test the accuracy of its position and to determine the point for the 
nail if required. 

Prolonging a Line. — It is required to prolong the line AB (Fig. 136) 
on to point C. If the distance AB is not too great and C is visible 
from A the theodolite may be set up at A, sighted on to B, and then 
points lined in ahead up to C. 

If the conditions of view require it, the theodolite should be set 
up at point B, the telescope sighted on to A and then transited so 
as to point ahead. If the instrument is in correct adjustment, any 
points which are now lined in will lie on the line AB produced. In 



142 SURVEYING 

prolonging a line by the above method errors of adjustment can be 
eliminated by proceeding as follows : Sight on A with the telescope 
in its normal position, transit the telescope and set out a point ahead. 
This point may fall at D (Fig. 136) a little to one side of the correct 
point C. Now sight on to A with the telescope in its inverted 
position and again transit the telescope. The same amount of 
error will be introduced as in the first case, but it will lie to the 
opposite hand, so that if a point E is now lined out the true point C 
will lie midway between D and E. The distance DE may be 
measured and point C fixed at the half distance. The theodolite 
may then be sighted on to C and intermediate points lined in 
between B and C. 

Another method of prolonging a line AB where both A and B are 
visible from C is to set up the theodolite as nearly at the point C as 
can be judged by the eye, sight on to point B, and note whether 
the centre of the cross hairs also strikes point A. If it does the 
theodolite is on the correct line. If it does not the theodolite must 
be shifted by trial till it is found to be in line. 

Ranging a Straight Line between Two Stations, neither of which is 
Visible from the Other. — This corresponds to the case described on 
p. 30, under chain surveying, where rising ground intervenes 
between the stations. It is assumed that both stations can be seen 
from some intermediate point. First employ the method of p. 30 
to line in two poles approximately by the eye. Set up the theodolite 
on the line thus found, sight to one of the stations, and transit the 
telescope. The line of sight should now strike somewhere near the 
other station. Note the amount of the deviation and estimate how 
far the theodolite should be shifted laterally in order to correct it. 
Shift the theodolite this estimated amount and test as before by 
sighting on one station and transiting the telescope. A few trials 
may be necessary before the theodolite is found to be in line. For 
accurate lining the final test should be made with the telescope first 
in the normal position and then in the inverted position. The 
theodolite having been brought into fine, intermediate points may 
now be ranged in. 



CHAPTER XI 

TRAVEKSE SURVEYING WITH THE THEODOLITE 

This chapter deals principally with the problems connected with 
the laying out of a system of traverse survey lines, the fixing of the 
survey stations, and the reading of the angles. The latter is an 
important part of the work involved in traverse surveying, and is 
dealt with somewhat fully. The various methods of procedure 
in use for measuring the direct angles between survey lines and for 
finding the whole circle bearings of survey fines are considered, 
and attention is given to desirable methods of booking angles and 
bearings. 

Traverse Surveying with the Theodolite. — In a traverse survey a 
system of connected survey fines is laid out from which the 
objects and natural 
features are located. 
The relative direc- 
tions of the fines are 
fixed by reading the 
angles which they 
make with each other, 
or by taking their Fig. 137.-Unclosed Traverse. 

compass bearings. 

The arrangement of the lines is not limited to any particular 
geometrical form as in chain surveying, where a system of 
triangles forms the fundamental basis of the arrangement. The 
use and limitations of the surveyor's compass for reading angles 
and bearings have been described in Chapter IX. 

Systems of traverse survey fines may be divided into the three 
following classes : — 

(a) Unclosed traverse. 

(b) Single closed traverse. 

(c) Network, consisting of combinations of (a) and (6). 

Fig. 137 represents an arrangement of survey lines forming an 





144 SURVEYING 

unclosed traverse. It corresponds to the case of a survey for a 
stretch of road where each successive straight portion is located 
from a single survey line. The directions of the lines are fixed by 

reading the angles at 
the bends. 

A single closed tra- 
verse is shown in Fig. 
138. The lines may 
form a polygon of any 
shape and with any 

number of sides. The 
Fig. 138. — Single Closed Traverse. .. , , 

survey lines to locate 

the boundaries of any single enclosure will usually take this form. 

The lines to survey any extensive area will usually take the form 

of a network, composed of a connected series of polygons with 

or without unclosed branches. This arrangement is typified in 

Fig. 139. 

Laying out Traverse Survey Lines. — The principles which should 
govern the lay-out of a system of traverse survey lines are in many 
respects similar to those 
which control the arrange- 
ment in a chain survey. 
The main difference is due 
to the fact that in traverse 
surveying the arrangement 
is not restricted to a par- 
ticular form, and in conse- 
quence the lines can be laid 
out to much better advan- 
tage as regards ease of loca- 
tion of objects. Time spent 

in examining the ground 

■ .i . • Fig. 139. — Traverse Network. 

to ensure that no serious 

obstacles to chaining occur on the survey lines will rarely be wasted. 

The aim should be to have a system which will be economical, both 

as regards field work and plotting, and to this end the method 

and requirements of plotting must be kept in view. For plotting 

purposes generally the fewer the number of lines the better, but 



I 




TRAVERSE SURVEYING WITH THEODOLITE 145 

this does not always hold good. An extra survey line may some- 
times be of great use in simplifying the booking and enabling 
complicated objects to be more easily and accurately located. 

Fixing Survey Stations. — While the survey stations are largely 
controlled by the conditions considered in the preceding paragraph, 
they must at the same time be chosen with due regard to the safety 
and ease and accuracy of manipulation of the theodolite. 

In open country firm, level ground for setting up on is to be 
desired. A position should be chosen where there is little likelihood 
of the peg being disturbed or removed, and near some permanent 
and easily recognised object to which it may be referred for easy 
recovery. 

In fixing stations on roads, streets or railways it is important to 
choose positions where the instrument will be safe and where the 
reading of angles can be carried on without interruption due to 
traffic. The setting up of the instrument near the centre of a road 
or street should be avoided. In streets it will generally be found 
most suitable to fix the stations on or near the kerb and have the 
lines run along the pavement or side of the road. In busy roads 
and streets an endeavour should be made to avoid positions where 
delay is likely to be caused through traffic. 

In surveying railway lines the stations should, as far as possible, 
be fixed quite clear of the running tracks, so that the observer and 
instrument will not be in danger from passing trains. 

Methods of Reading Angles. — With a theodolite whose circle is 
graduated from 0° to 360°, either of the following two methods of 
reading angles in a traverse survey may be employed : — ■ 
(a) Direct angle or separate angle method. 
(6) Whole circle bearing method. 
If the theodolite circle is graduated according to the quadrant 
system, the deflection angle method of reading angles may be 
employed. The quadrant method of graduation is rarely used in 
Britain. 

Direct Angle or Separate Angle Method. — The direct angle method 
consists in obtaining the actual angle at the junction at each pair of 
lines. The angle read may either be the interior or the exterior 
angle, as shown in Fig. 140. In reading a series of angles, however, 

s. L 



146 



SURVEYING 



a systematic method of procedure should be adopted. Interior and 
exterior angles should not be read indiscriminately. 

In an unclosed traverse, as illustrated in Fig. 137, assuming that 
the stations are occupied in the order of their numbers, the method 

of procedure would be as follows : 
The theodolite would be set up 
at station 2 and sighted on 
station 1, with the vernier set at 
zero. If the theodolite is pro- 
vided with a compass the bearing 
of line 2 — 1 should be noted as 
an approximate method of deter- 
The upper clamp would then be loosened and 

The angle 




Fig. 140.— Beading Exterior or 
Interior Angles. 



mining true north 

the telescope rotated clockwise and sighted on station 3 
which would be read off the circle is shown by the arrow in the figure, 
On setting up at any other station the telescope would be sighted 
back on the previous station with the vernier set at zero and then 
turned on to the for- 
ward station. Pro- 
ceeding in this 
manner, the angles 
read would all lie on 
the same side of the 
traverse, the calcula- 
tion of the bearings 
or azimuth angles of 
the lines being there- 
by rendered simple. 
Angles read by the 
direct method may 
be checked by doub- 
ling, while, if a more 
precise value i 
desired than can be 




s Fig. 141.- 



-Calculation of Bearings from Separate 
Angles. 



got from a single observation, the method of repetition may be 
employed. See Chapter XIV. 



Calculation of Bearings from Direct Angles. — In a closed traverse 
it is generally preferable to read the internal angles. Fig. 141 



TRAVERSE SURVEYING WITH THEODOLITE 147 

shows a closed traverse in which the angles read are indicated by 
small arcs with arrow heads showing the direction of rotation of 
the theodolite. The observed compass bearing of the line AE 
is 25° west of magnetic north and the angle EAB is 120°. The 
whole circle bearing of the line AB is therefore 120° — 25° = 95°. 
Consider now the angles at the point B. The bearing of the line BF, 
which is AB produced, is 95°. A pointer hinged at B and pointed 
in this direction would require to be rotated in a clockwise direction 
through 180° + 100° to bring it round to the direction BC. The 
whole circle bearing of BC is, therefore, 95° + 180° + 100° = 375°. 
As this value is greater than a complete revolution, 360° must be 
deducted from it, so that the proper bearing is 375° — 360° = 15°. 
In the same way the bearing of CD is got by adding 180° -f 45° to 
the bearing of BC, and so on for the other lines. The bearings 
worked out for the whole polygon are as shown in the following 
table : — 



Line. 






Bearing. 








AB 


120° 


- 25° 








95° 


BC 


95° 


+ 180° 


+ 100° 


= 375° 


= 


15° 


CD 


15° 


+ 180° 


+ 45° 


= 240° 


= 


240° 


DE 


240° 


+ 180° 


+ 210° 


= 630° 


= 


270° 


EA 


270° 


+ 180° 


+ 65° 


= 515° 


= 


155° 



In reading angles by the direct angle or separate angle method, 
as above described, each angle is dealt with independently of any 
other. The angles may, therefore, be read in any order, and this is 
sometimes an advantage as compared with the whole circle bearing 
method to be next described. 

When all the interior angles of a polygon have been read a check 
on their accuracy is obtained by adding them together and noting 
whether their sum amounts to an even number of right angles. In 
a polygon of N sides the sum of the interior angles should be equal 
to 2N — 4 right angles, or equal to 180 (N — 2)°. 

If the angles are read in sequence round the polygon, and the 
bearing of each line is calculated as the work proceeds, a rough 
check will be afforded by comparing the calculated bearing with 
the compass bearing. 

l2 



148 SURVEYING 

Whole Circle Bearing Method. — In this method, known also as 
the azimuth angle method, a reference direction, sometimes known 
as a reference meridian, is chosen, and the direction of any given 
line is determined by measuring the clockwise angle between this 
reference direction and the given line. The angle will lie between 
0° and 360°. The reference meridian is generally taken as magnetic 
north or true north, but any other direction, such as that of a 
particular survey line, may be adopted if more convenient. 

The three following methods of procedure are used for the purpose 
of finding consecutively the whole circle bearings of the lines of a 
traverse survey : — - 

(a) Direct bearing method. 

(b) Back bearing method. 

(c) Method by ignoring distinction between forward bearing 
and back bearing. 

Method (a). — The procedure applied to the determining of the 
whole circle bearings of the two lines AB and BC by the direct 
bearing method is illustrated in Fig. 142. Magnetic north is 
assumed to be the reference direction. The instrument is set up 
and levelled at station A. The vernier of the horizontal circle is 
set to zero and clamped, the lower clamp being meantime loose. 
The magnetic needle having been lowered on to its pivot, the plates 
and telescope are rotated together till the zero of the compass 
circle is brought to coincide with the north-pointing end of the 
needle. The telescope then points in the same direction as the 
needle, that is, towards magnetic north, and the index of the 
horizontal circle is still at zero, or, in other words, the telescope now 
points in the reference direction and the reading is 0°. The lower 
clamp is now set, thus fixing the horizontal circle, and the upper 
clamp is loosened. The telescope is then rotated clockwise and 
directed on to station B and the angle is read off. This angle is 
the whole circle bearing of the line AB with reference to magnetic 
north. In the small diagram (A) (Fig. 142) the full lines represent 
the telescope pointing to magnetic north while the circle reading 
is zero. The dotted lines represent the position of the telescope 
when pointing towards B. The lower clamp is loosened, and the 
instrument may then be lifted and transported to station B and set 
up, the plates remaining clamped at the bearing of AB. The 



TRAVERSE SURVEYING WITH THEODOLITE 149 



instrument having been set up and levelled, the telescope is sighted 
back on station A, the lower clamp is fixed, and the cross hairs are 
accurately set on A by turning the lower tangent screw. The 
position of the telescope and horizontal circle is now as shown in 
full lines on small diagram (B) (Fig. 142). It will be noted that the 
diameter of the circle which pointed to magnetic north at station A 
points in the same direction at station B, but turned through 180°. 
The telescope is now transited, so that if it was normal at A it will 
now be inverted, and the vernier will still be reading the bearing 




Angle Read 500° 
(C) 



(B) 



Telescope Inverted 
Angle Read GO" 



Fig. 142. — Whole Circle Bearings by Direct Bearing Method. 

of hue AB. The transiting of the telescope cancels the effect of the 
rotation through 180° of the horizontal circle which has taken place. 
Now loosen the upper clamp and sight the telescope on to station C 
and read the angle, which will be the forward whole circle bearing 
of the line BC. 

The procedure at any other station such as C is similar to that 
at B, and may be summarised as follows :— 

(a) Set up at C with vernier clamped to bearing of BC and the 
telescope in the position used for reading that bearing. Lower 
clamp loose. 

(b) Sight telescope back on station B, using lower clamp and tan- 
gent screw for accurate adjustment. 



150 SURVEYING 

(c) Transit the telescope, loosen the upper clamp, and sight the 
telescope on station D, using the upper clamp and tangent screw 
for accurate setting. 

(d) Read off the angle which will be the whole circle bearing of 
the line CD. 

A reference to Diagrams (A), (B) and (C) (Fig. 142), should make 
the procedure clear and show how the angles read are the whole 
circle bearings of the various lines. If the theodolite has more than 
one vernier the same vernier must be used throughout. 

The method of procedure above described, in which each angle is 
read only once, furnishes in itself no check on the accuracy 
of the angles. Comparison of the bearings with the readings 
of the compass (if there is one) will give a very rough check. 
To obtain greater reliability proceed as follows at station B after 
having read the bearing of the line BC : Loosen the upper clamp, 
rotate the telescope clockwise and sight back on to station A. The 
circle reading should now be equal to the forward bearing of the 
line AB increased or diminished by 180°. If there is no discrepancy, 
rotate the telescope clockwise and again direct it on to station C, 
and read the bearing. If this is the same as before, the angle may 
be accepted as correct. The upper clamp would, therefore, be set 
at this reading, the lower clamp would then be loosened and the 
theodolite could be transported to the next station. 

If on sighting back on to station A a small discrepancy is found 
in the bearing, it will be necessary to repeat the operation. 

Where the bearings of several lines require to be read from one 
station, read first the bearings of all the lines, taking them in order 
in a clockwise direction, and then check back on to the starting 
station. Finally, sight the telescope on to the station at which the 
next set-up is to take place, see that the bearing agrees with the 
former reading, clamp the plates at this bearing and carry the 
instrument forward. 

Method (b). — In the back bearing method of procedure the 
telescope is always kept in its normal position ; it is never transited. 
Having set up the instrument at B set the vernier to the whole 
circle bearing of BA, which is the forward bearing of AB increased 
by 180°. Sight on to station A and then, with the lower clamp 
fixed, rotate the telescope clockwise and sight on station C. The 



TRAVERSE SURVEYING WITH THEODOLITE 151 

reading will be the whole circle bearing of the line BC. To ensure 
accuracy check back on to the starting station and repeat the 
observation. 

Method (c). — In this method the distinction between forward - 
bearing and back bearing is ignored. On setting up at B the 
telescope is sighted on to station A with the vernier set to the 
forward bearing of line AB. The bearing determined for BC will, 
therefore, be wrong by 180°. This will, however, be the correct 
backward bearing of CB, so that on setting up at C and proceeding 
as before the bearing of CD will be correctly determined. The 
bearings of alternate lines will in fact be correct while the others will 
be wrong by 180°. The only mistake in plotting which can arise 
thereby is the laying off of one survey line to the wrong hand with 
respect to another, as in the direction BC instead of BC (Fig. 142). 
Such a mistake will not occur if a sketch has been made showing the 
survey lines in fair relationship to each other, and in any case would 
not remain undetected in a closed traverse. 

Comparison of Methods of Measuring Whole Circle Bearings. — In 

the first or direct bearing method the vernier clamp may slip, due 
to jolting in carrying the instrument from one station to another, 
and hence the circle reading should be checked after the instrument 
is set up. If the instrument is not in correct adjustment in respect 
that the line of sight of the telescope is not exactly perpendicular to 
its horizontal axis, or if its horizontal axis is not in correct adjust- 
ment, an error will be introduced at each angle, due to transiting 
the telescope. This source of error is avoided in methods (6) and (c). 
Method (b), or the back bearing method, has the disadvantage that 
the back bearings of the lines require to be calculated and that the 
vernier requires to be set to a particular reading at each station. 
Both operations afford opportunity for the introduction of error 
and take up time. Method (c) avoids the necessity for calculation 
of bearings, but the vernier is liable to slip. 

Booking Angles of Traverse Survey. — Where the arrangement of 
survey lines is not complicated the most convenient method of 
recording the angles is to figure them directly on a sketch of the lines. 
A fair sketch of the fines with all stations lettered or numbered and 
with the length and bearing of each line plainly figured alongside is 



152 



SURVEYING 



a necessity if the survey lines are to be expeditiously plotted. 
Wherever the method of taking the bearings involves calculation 
and where the sketch method is not convenient, the following 
tabular system of booking the angles is recommended : — 



Theodolite Station. 


Station observed. 


A 


B = 335° 40' 


B 


A = 155° 40' 




L = 191° 13' 




C = 250° 4' 




K = 63° 35' 


C 


B = 70° 4' 




D = 293° 15' 




M = 342° 45' 


D 


C = 113° 15' 




E = 197° 20' 



The above system keeps a record of the back sight and its 
bearing taken at each station as well as the bearings of the 
several lines read from it. The back sight from C to station B 
is set down as 70° 4'. This is obtained by calculation from the 
bearing of BC previously found. The figures being set down in the 
notebook enables a mistake in calculation to be afterwards found 
out by checking and corrected, while, if the calculation is merely 
done mentally and not recorded, the detection of an error is a 
difficult matter. 



CHAPTER XII 

PLOTTING A TRAVERSE SURVEY BY ANGLE AND DISTANCE 

There are two principal methods of plotting the survey lines of 
a traverse survey, viz., by angle and distance and by co-ordinates. 
The various ways in which the angle and distance method of plotting 
a system of traverse survey lines may be applied are considered in 
this chapter. The plotting of angles and bearings may be under- 
taken by methods which correspond to the direct angle and whole 
circle bearing methods of measuring angles, with variations in each 
case according as the angles are laid off by protractor or by geo- 
metrical construction. The methods of laying-off angles and bear- 
ings by the protractor and by geometrical constructions, utilising 
the tangent or chord of the angle, are dealt with, and the advantages 
of the various methods are compared. Consideration is also given 
to the methods of checking and ensuring the accuracy of unclosed 
traverses, and to the adjustment of the closing error in a closed 
traverse. 

The angle and distance methods of plotting traverse surveys 
described in this chapter are only suitable for small surveys, and are 
much inferior, in respect of accuracy of plotting, to the co-ordinate 
method described in the next chapter. 



Methods of Plotting. — In laying down the directions of traverse 
survey lines on paper for the purpose of plotting the plan we may 
proceed according to either of the two general methods already 
mentioned, viz. : — 

(a) Direct or separate angle method. 

(b) Whole circle bearing method. 

(a) The direction of each survey line is laid off from the previously 
plotted line to which it is connected by plotting on the paper the 
angle at the junction of the lines. 

(b) The directions of the lines are laid down relative to a fixed 
reference direction. 



154 SURVEYING 

Of the two methods the latter is the more useful and reliable. 

Whichever is used the separate angles and bearings may be 
plotted by protractor or by geometrical construction in several 
different ways, the following methods being noteworthy : — 

(a) By protractor. 

(b) Tangent method. 

(c) Chord method. 

Methods (a) and (b) are the more useful for purposes of plotting 
a traverse survey by angle and distance. 

Laying off a Single Angle or Bearing. — (a) By protractor. To 
lay off a given angle at point B on the base AB, first produce the 
line AB onwards by a length rather greater than the radius of the 
protractor. Then set the 0° and 180° points of the protractor on 
this line and bring the centre mark to coincide exactly with the 
point B. Make a prick mark opposite the proper graduation on the 
circumference and join it up to point B. 

The usual form of semicircular protractor of brass or other 
material is graduated from 0° to 180° in both directions, so that 
either clockwise or anti-clockwise angles may be set off. The 
accuracy of the work is limited by the size of the radius, and as 
this does not commonly exceed 6 ins., such a protractor is only 
useful for plotting short lines and details. 

By using a full-circle protractor and pricking off both the given 
angle and its supplement, the plotted side of the angle will be fixed 
by two points a whole diameter apart, and hence greater accuracy 
will be attainable than when the points are only a radius apart. 

A good form of protractor for plotting survey lines is the large 
circular cardboard type, 18 ins. to 24 ins. in diameter. The cir- 
cumference is graduated to ten-minute intervals, and these are 
large enough to permit of single minutes being pricked off by 
estimation with an error not exceeding two or three minutes. For 
plotting whole circle bearings there are two sets of graduation 
figures running both in clockwise direction from 0° to 360°. The 
one set follows the other with an interval of 180°, so that any given 
angle occurs twice at points diametrically opposite each other. 
In using this protractor both points are pricked off, thus giving a 
base equal to the whole diameter. Directions so pricked off are 
transferred and made to pass through the required station point 



PLOTTING BY ANGLE AND DISTANCE 



155 




on the plan by the use of a parallel ruler or other equivalent 
means. The centre of the protractor does not in general require 
to be used in setting, or in laying off angles. 

(b) By tangent methods. In the right-angled triangle (Fig. 143), 
if the base is one unit long and the length of the perpendicular is t, 

the tangent of the angle a will be y = t. Any angle can, therefore, 

be laid off by erecting on a base line one 

unit long a perpendicular equal in length to 

the natural tangent of the angle. If the 

base is made a certain number of units in 

length the perpendicular will require to be , Unjt 

made equal to the tangent multiplied by _p IG . 143. _ plotting 

that number. For many purposes a con- Angle by Tangent 

venient length of base is 10 ins., and in e ° 

that case the height of the perpendicular in inches is ten times the 

natural tangent. 

For angles less than 45° the length of the perpendicular is less than 
that of the base. For angles greater than 45° the perpendicular 
is greater than the base, and as the angle approaches 90° the 
perpendicular becomes immensely greater. It 
is, therefore, not convenient to plot angles 
which are much greater than 45° directly by the 
tangent method. For such angles it is better 
to plot the complement of the angle from a 
base drawn perpendicular to the given side. In 
Fig. 144 the plotting of the angle a on the side 
ab requires the erection of the long perpendicular 
be. This is avoided by plotting the angle /3 from 
the base ad drawn at right angles to the line 
ab. If ad is 10 ins. long de = 10 tan /3 = 10 tan 
(90 — a). 
Fig. 145 shows how the direction of any 
whole circle bearing between 0° and 360° can be laid off by 
utilising the sides of four squares as bases. A little consideration 
will show that one square can quite well serve the purpose of the 
four. 

The accuracy of the tangent method of plotting angles depends on 
the precision with which distances are scaled and right angles laid off. 




Fig. 144.— Plotting 
Angles over 45°. 



156 



SURVEYING 



Particular care should be taken to see that the set-squares used are 
accurate. The accuracy of base lines laid out in the form of a 
square is best tested by scaling the lengths of the diagonals and 
noting if they are the same. 

(c) By chord method. It is required to lay off an angle a with 




90 270' 



360' 



Bearings 
90° - 180' 



IW 270° 
Bearings Bearings 



180'- 270° 270'- 360° 

Fig. 145. — Plotting Bearings by Tangent Method. 

apex at a on the line ac (Fig. 146). With a as centre and radius ab 
equal to a convenient number of units, say, ten, an arc bd is struck. 
From centre b and with radius equal to the chord corresponding to 
angle a for a radius ab another arc is struck to intersect the former 
in point d. The required angle dab is got by joining point d to 
point a. The lengths of chords of angles corresponding to unit 
radius are given in various sets of 
mathematical tables. The chord bd 
corresponding to the angle a and unit 
a 
2' 




radius ab is equal to 



Fig. 146 —Plotting Angle by Using the protractor. 
Chord Method. , ,-, -, • 

trates the direct or 



Direct or Separate Angle Method of 

Plotting Traverse Survey Lines. — (a) 

Fig. 147 illus- 

separate angle 

method of plotting survey lines by protractor. Before commencing 
to lay down any lines on the paper, precautions must be taken to 
ensure that the survey lines when plotted will lie in proper position 
on the sheet. See Chapter VIII. 

In Fig. 147 the line AB is taken as the base. The direction of 
the line AE is got by placing the protractor as shown, and making a 
prick mark at the 71° graduation. The point E lies on the line 
drawn through this prick mark and point A, and its position is 



PLOTTING BY ANGLE AND DISTANCE 



157 



fixed by scaling off a distance of 707 ft. from point A. Lines BC 
and CD are laid off successively in a similar manner to the line AE. 




Fig. 147. — Plotting Survey Lines by Protractor. 

The angle of 223° at point D is plotted by laying off an angle of 
43° above the line CD produced. The last line drawn from D 
should pass through the point E already plotted, and the scaled 
length of DE should 



be the same as the 


••,. 










actual length 




y£ 






measured on the 




_^o\_ 






ground. 




70 N 






If the final plotted 




10 


f\£i_^ 


^ / 


point E' does not c 
coincide with the 


'$44/9 






c 


■vS 


first point E the 
interval EE' is 


<a / 






to 
to 




known as the closing 
error. The closing 


Ar° 




I / 10 


Oi 




error of a traverse 
plotted with a pro- 
tractor will, in 


4 

Fig. 148 


B 
— Plotting Survey Lines by 
Method. 


Tat 


lgent 



general, be due to a combination of errors in surveying and errors 
in plotting. If the closing error turns out large, mistakes should at 
once be looked for in the plotting. To test for angular errors, 
check first the angle which the direction of the last line makes with 



158 



SURVEYING 



the first line. If this does not agree with the measured angle, check 
the others till the mistake is located. If the angles are found to 
be correct the error must be looked for in the scaling. 

The distances and angles of an unclosed traverse require very 
careful checking to ensure that no errors have occurred in the 
plotting, as the work affords no check in itself, such as is obtained 
on completing a closed traverse. 

(6) Tangent method. Fig. 148 illustrates the tangent method 
of laying off angles applied to the plotting of traverse survey lines 
successively. The first step is to make out a table showing the 
angles to be plotted with the values of their tangents and the lengths 
of the survey lines as follows : — 



Angle plotted at 


Tangent. 


Line. 


Length. 


A = 


90° — 71° = 19° 


0-3443 


AB 


575' 


B = 


180° — 136° = 44° 


0-9657 


BC 


660' 


C 


= 40° 


0-8391 


CD 


515' 


D = 


223° — 180° = 43° 


0-9325 


DE 


393' 


E = 


90° — 70° = 20° 


0-3640 


EA 


707' 



The line AE, making an angle of 71° with the base AB, is got by 
laying off an angle of 19° on the right-hand side of a line drawn up 
at right angles to AB from the point A. The angle of 19° is plotted 
by marking off the length A/ equal to ten units and from / con- 
structing the offset /<? equal to ten times the tangent of 19° or equal 
to 3-443 units. The units on any engineer's decimally-divided 
scale may be used instead of inches if a length of base other than 
10 ins. is thought desirable. The line Kg makes the required angle 
with the line AB, and the point E is plotted by scaling off a length 
of 707 ft. from A. The direction of the line BC is got by marking 
off from B a base of ten units along the line AB produced, and 
erecting an offset of 9-66 units at its right extremity. The fines 
CD and DE are plotted successively in the manner shown in the 
figure. If the traverse does not close, the angle made by the last line 
drawn from D with the line AE should be checked. If no error is 
found in the angle the lengths of the traverse lines must be checked. 

Whole Circle Bearing Method of Plotting Traverse Survey Lines.— 

(a) By protractor. This method is illustrated in Fig. 149, and with 



PLOTTING BY ANGLE AND DISTANCE 



159 



careful draughtsmanship and the use of a large and accurate pro- 
tractor enables traverse survey lines to be laid down with fair 
accuracy within, at least, the limits of a double-elephant sheet. 
A line parallel to the base AB, fg in the figure, is drawn in a con- 
venient central position on the sheet and the protractor is placed 
over this line so that the graduations representing the bearing of 
AB are coincident with it. The protractor is thereby set with its 
zero and 180° points on the reference meridian, and the bearings 
of the other lines may be pricked off round its circumference and 




Fig. 149. — Plotting Whole Circle Bearings by Protractor. 

marked for identification in the manner shown. Instead of marking 
the bearing by the line to which it refers, its value may be 
written on the paper at each prick mark. To lay off the direc- 
tion AE through point A, set a parallel ruler with its edge 
passing through the two prick marks designated AE. Then roll 
it till the edge passes through point A and draw a fine pencil 
line through it in the proper direction. Point E is plotted by 
scaling along this line from A. The other lines are similarly laid off 
by transferring the directions with the parallel ruler, and the 
accuracy of the work is proved if the last line drawn through D 
passes through point E and scales correctly. This method of 



160 



SURVEYING 



plotting requires that the parallel ruler should be very carefully 
used to ensure accurate results. It should be carefully tested to 
see that it rolls true by the method described on p. 87. A good check 
against mistakes in marking off the bearings is got by pricking the 
centre point of the protractor on to the paper and, as the edge of 
the ruler is laid across each pair of points, noting whether it also 
passes through the centre point. A mistake in marking off one of 
the points from the protractor will thereby be at once detected, as 
the line joining the points would not then pass through the centre. 

As compared with the two previous methods of plotting traverses 
the whole circle bearing method has the advantage that one setting 
of the protractor serves to lay down all the bearings. Also, while 
in the previous methods a small error in the direction of any one 
line affects the direction of all subsequent lines connected thereto, 
in the whole circle bearing method each angle is plotted indepen- 
dently of any other, and a small error in the bearing of one line 
does not affect the direction of any other line. The method has 
the further advantage that the full diameter of the protractor is 
utilised in laying off the angular directions. 

(6) By tangent method. This method of plotting whole circle 
bearings is illustrated in Fig. 150. The bearings of the survey 
lines require first to be reduced to the angles less than 45° which 
they make with the meridian direction or with a direction at right 
angles to the meridian. The meridian direction is represented by 
the bearings 0° and 180°, and the other direction by the bearings 
90° and 270°. The following table shows the reduced angles for 
the given traverse, and the tangents required in plotting : — 



Line. 


Length. 


Bean 


ng. 


Angle Plotted. 


Tangfiit. 


AB 


575 


90° 


0' 




0° 


•0000 


BG 


660 


46° 


0' 


90° — 46° - 


44° 


•9657 


CD 


515 


266° 


0' 


270° — 266° = 


4° 


•0699 


DE 


393 


309° 


0' 


309° — 270° = 


39° 


•8098 


EA 


707 


199° 


0' 


199° — 180° = 


19° 


•3443 



A square is constructed near the centre of the sheet of paper, 
having sides long enough to form suitable bases for the plotting 
of the angles. Sides 10 or 20 ins. long may be used, according to 



PLOTTING BY ANGLE AND DISTANCE 



161 



the length of the survey lines. The square will not necessarily 
be placed with its sides parallel to the edges of the paper, but must 
be arranged to suit the desired disposition of the survey lines. 
Having fixed one of the survey lines on the paper, plot from this 
line by the tangent method either the meridian direction or the 
direction perpendicular to it and make the sides of the square 
parallel to these. Whole circle bearings between 0° and 45° will 
be plotted by making gf the base and laying off the offsets along/&. 
For bearings between 45° and 90° gh is the base and offsets are 




k~3-44 



t - -- 10 ->j 

Fig. 150.— Plotting Whole Circle Bearings by Tangent Method. 

erected along hk. For bearings between 90° and 135° fk is the base 
and offsets are plotted down the line kh, and so on. In plotting the 
bearing of line EA, which is 199°, the line kh represents a bearing 
of 180° and the additional angle, namely, 19°, is plotted on hg as 
base. If kh is 10 ins. long the offset hi will be 3-443 ins. long, and 
the line kl will give the direction of AE. The directions of the 
other lines are laid off in similar manner. The plotting of the 
survey lines is effected by transferring these directions by parallel 
ruler or other means, the procedure being the same as when the 
angles are plotted by protractor. 



162 SURVEYING 

Comparison of Angle and Distance -Methods. — The direct angle 
method of plotting survey lines, in which the direction of each line 
is laid off relative to that of the previously plotted adjoining line, 
is not recommended except for rough purposes. Angular error 
occurring at any point is continued through the rest of the work, 
and the method is laborious in respect of the large number of 
separate settings of the protractor, or of the separate geometrical 
constructions required in order to lay off the directions of the 
various lines. 

Where the angles of a traverse have been measured by the direct 
angle method, the preferable procedure is to reduce them by cal- 
culation to whole circle bearings in the manner shown on p. 147, 
and then adopt one of the corresponding methods of plotting. 
Of the angle and distance methods of plotting a traverse, the 

whole circle bear- 
D ing method using 

a large and accu- 
rate protractor is 
the most generally 
useful. The tan- 
gent method, with 
the use of base 
lines of suitable 



Fig. 151. — Method of Checking an Unclosed Travers 



length, will give as accurate results as the protractor method, but 
it is somewhat more laborious and affords more opportunity for 
mistakes occurring in the reduction of bearings and abstracting of 
trigonometrical values from the tables. It is, at the same time, a 
most useful method to have at command should the occasion arise 
when a protractor is not available. 

Checks on Unclosed Traverse.— A method of ensuring accuracy 
in the bearings of an unclosed traverse is illustrated in Fig. 151. 
From station A, in addition to reading the bearing of the first 
line AB, take also, if reasonably possible, the bearings to one or 
more of the forward stations, such as D. On leaving station A the 
theodolite would be set up successively at B and C to get the 
bearings of the lines BC and CD. On setting up at D the back 
sight should be taken on to station A, as giving a longer sight and 
a more directly obtained bearing than station C, and, after taking 



PLOTTING BY ANGLE AND DISTANCE 163 

the bearing of DE and check bearings where possible to stations 
ahead, read the bearing of DC. If this corresponds with the bearing 
of the same line taken from C, the angular work up to station D 
may be accepted as correct. If there is a small discrepancy not 
exceeding the unavoidable error for the instrument used, the 
bearing of DC taken from station D will be accepted as correct, 
the bearings of BC and CD being adjusted slightly to get rid of the 
angular closing error. By proceeding in this way the surveyor will, 
under favourable conditions, obtain a check on the angular work 
every few stations and will be enabled to leave the field confident 
of its accuracy. 

By making use of the observed check bearings in the plotting a 
partial check on the accuracy of the linear work may also be 
obtained. If the bearing of AD drawn through A is found to pass 



Fig. 152. — Checking Unclosed Traverse by Bearings to 
Lateral Object. 

through point D, the lengths may be presumed to have been cor- 
rectly taken and plotted as well as the angles. An error in measur- 
ing or plotting the length of the lines AB or CD would evidently 
cause point D to deviate from the line of the bearing drawn through 
A. If, however, any line, such as BC, had the same, or nearly the 
same, bearing as AD, an error in its plotted length due to measure- 
ment or scaling would not cause point D to deviate from the line AD, 
and hence would not be detected. 

A method which furnishes a check when the work is plotted 
consists in reading the bearings to a conspicuous side object from 
each of three or more consecutive stations, the whole traverse being 
dealt with by a series of such groups of stations. In Fig. 152 side 
bearings have been taken from stations A, B, C, and D to the 
point P. The check in plotting consists in laying off these bearings 
through their respective stations on the paper and noting whether 
the fines pass through one point. If they do, the work is presumably 

m2 



164 SURVEYING 

accurate. The method is not a very satisfactory one. The angles 
are not checked in the field and the office check is often awkward 
owing to the intersection points of the side bearings falling off the 
paper. 

For important work the most satisfactory method of checking the 
linear measurements consists in chaining each survey line a second 
time, preferably in the reverse direction. 

Graphical Adjustment of Closing Error. — A graphical method of 
adjusting the unavoidable closing error in a traverse plotted by 
any of the foregoing methods is illustrated in Fig. 153. The 




Fig. 153. — Graphical Adjustment of Closiug Error. 

starting point of the traverse is E and the last line drawn from 
D terminates at E', giving a closing error EE\ To eliminate the 
closing error E' must be moved to E, and, to avoid making a large 
adjustment in any single line, each of the other station points 
should be moved a certain distance parallel to E'E, the amount of 
the shift for each point being proportional to its total distance 
from the starting point E. The amount of the shift for the various 
station points is obtained graphically, as shown in Fig. 154. The 
lengths of the survey lines starting from point E are laid off to a 
convenient scale along the line EABCDE', and the perpendicular 
E'E" is constructed equal in length to the closing error on the paper. 
The line joining E to E" will by the cut-off of the perpendiculars 



PLOTTING BY ANGLE AND DISTANCE 



165 



erected at A, B, C, and D give the required amounts of the shifts 
at these stations. The various stations being shifted parallel to 
the direction EE', as shown in Fig. 153, by the amounts thus 



E A B C D £' 

Fig. 154. — Distribution of Error in Traverse. 

obtained we get the adjusted form of the traverse EabcdE. It will 
be noticed that by this method of adjustment the total error is 
distributed throughout the traverse, so that each line receives only 
a small linear and angular alteration. 



CHAPTER XIII 

PLOTTING TRAVERSE SURVEY BY CO-ORDINATE OR LATITUDE AND 
DEPARTURE METHOD 

This chapter deals with the methods of calculating the latitudes 
and departures of traverse survey lines, the finding therefrom of the 
co-ordinate distances of the stations referred to two co-ordinate 
axes, and the plotting of the survey stations by means of these 
co-ordinates. It deals also with the adjustment of angular errors 
and closing errors, and consideration is given to the problems 
involved in altering the bearings of survey lines to suit new co- 
ordinate axes and in connecting survey lines from one sheet to 
another. Finally, the significance of closing error is considered 
and limits of permissible closing error under various conditions 
are given. 

Co-ordinate Method of Plotting Traverse Survey. — The co-ordinate 
method is the most accurate and satisfactory method of plotting a 
traverse. It enables the closing error of the field work to be 
accurately determined before plotting is commenced and to be 
adjusted if necessary, whereas with the angle and distance methods 
the closing error is not found till the lines are plotted, and even then 
it is not known how much of the error is due to the field work and 
how much has arisen in the plotting. Further, in the co-ordinate 
method each station point is plotted separately and independently, 
so that errors due to draughtsmanship are not accumulated. Where 
reliable results are desired, the labour in working out the co-ordinates 
by the methods explained in the following pages is far outweighed 
by the satisfaction which the surveyor has in the proved consistency 
of his field work and the confidence that none of the work expended 
in plotting will be wasted. 

The co-ordinate method is by far the best to adopt where the work 
is of such extent as to require to be plotted on separate sheets. 
When the co-ordinates have been worked out the plotting of the 
survey can proceed on several sheets at the same time. 



PLOTTING BY CO-OKDINATE METHOD. 



167 



Latitude and Departure. — The term " latitude " denotes the co- 
ordinate length of a survey line measured parallel to an assumed 
meridian direction which may be either true north, magnetic north, 
or any reference direction which may suit the purpose. The term 
"departure " denotes the co-ordinate length of a survey line measured 
at right angles to the meridian direction. In Fig. 155 the length 
AD measured north and south is the latitude of AC, while AB 
measured east and west is its departure. Similarly the lengths 
CH and CG are the latitude and departure respectively of the survey 
lineCF. 

The magnitudes of the latitude and departure of a survey line are 



N 
D 

.■Si 


%H j 

CVy^_ Departure 

■&£/ 

/Departure \ 


c ..Departure_ 
G 


A 


B 1 
S 


< E 



Fig. 155. — Latitudes and Departures of Survey Lines. 

found by the principles of trigonometry. If the line AC (Fig. 155) 
makes the angle a with AN then — 

Latitude AD = AC cos a 
Departure AB = AC sin a 

Similarly if line CF makes an angle y8 with the meridian direction 

Latitude CH = CF cos £ 
Departure CG = CF sin j3 

Co-ordinates. — The latitudes and departures of the survey lines 
of a traverse are calculated in order to determine the co-ordinate 
distances of the stations from two axes. If a north and south line 
has been adopted as the reference direction for the bearings the 
co-ordinate axes for plotting purposes will in general be taken in the 



168 



SURVEYING 



north and south and east and west directions. For convenience 
also one or both of the axes will usually be taken through one of the 
survey stations. In the figure AN and AE are the co-ordinate 
axes. The co-ordinates of point C are AB and BC equal respectively 
to the departure and latitude of the line AC. The co-ordinates of 
point F are AK and KF. AK is equal to the sum of the departures 
of the lines AC and CF, while KF is equal to the sum of their lati- 
tudes. The co-ordinate of point L along the axis AE will be obtained 
by adding the departure of the line FL to the co-ordinate AK of 
point F, while its perpendicular co-ordinate will be obtained by 
subtracting the latitude of FL from FK. The determination of 
the co-ordinates of the survey points along one axis is thus accom- 
plished by the successive addition and subtraction of departures, 




Fig. 156. — Calculation of Latitudes and Departures from Whole Circle 
Bearings. 

while the co-ordinates along the other axis are obtained by the addi- 
tion and subtraction of the latitudes. 

Distinction between North Latitude and South Latitude and between 
East Departure and West Departure. — In summing up latitudes and 
departures to obtain the co-ordinates of the stations the surveyor 
must consider that he is proceeding along the survey lines in suc- 
cession one way round. Then any line traversed over in a north- 
easterly direction will be considered to have a north latitude and 
an east departure. If the direction of motion in passing over a 
line is towards S.E. the line will have south' latitude and east 
departure. Direction towards S.W. means south latitude and west 
departure, while N.W. direction means north latitude and west 
departure. North latitudes and east departures are considered 
positive and south latitudes and west departures negative. The 
summations to arrive at the co-ordinates are then made algebraically. 

The diagrams in Fig. 156 and the accompanying table illustrate 



PLOTTING BY CO-ORDINATE METHOD. 



169 



the calculation of latitudes and departures from whole circle bearings, 
and show the proper signs for bearings which occur in each of the 
four quadrants. The whole circle bearing is represented in each 
case by a and the reduced bearing required for calculation purposes 
is represented by /3. 

Calculation of Latitudes and Departures from Whole 
Circle Bearings. 





Quadrant. 


Reduced 
Bearing. 


Latitude. 


Departure. 




Value. 


Sign. 


Value. 


Sign. 


AB 
CD 
EF 
GH 


N.E. 
S.E. 
S.W. 
N.W. 


0= ]80°— a 
/3 = a — 180° 

p = 360° — a 


AB cos p 
CD cos p 
EF cos p 
GH cos p 


N(+) 

S(-) 
S(-) 
N(+) 


AB sin jS 
CD sin/3 
EF sin/3 
GHsin/3 


E (+) 
E (+) 
W(-) 
W(-) 



It will be seen that the reduced bearing required for the computa- 
tions is the angle (less than 90°) which the survey line makes with 
the meridian direction. For a bearing in the N.E. quadrant no 
reduction is required. When the whole circle bearing lies between 
90° and 180° subtract it from 180° to obtain the reduced bearing. 
When it lies between 180° and 270° subtract 180° from it, and when 
it lies between 270° and 360° subtract it from 360°. 

Bearings taken with a compass graduated on the quadrant 
system with zeros at the north and south points require no reduction. 

When the angles of the traverse have been calculated by the direct 
or separate angle method the whole circle bearings of the survey 
lines must first be calculated in the manner shown on p. 147, and 
the reduced bearings can then be worked out. 

Calculation of Latitudes and Departures. — The method of multi- 
plying the length of the survey line by the natural sine and cosine 
of the reduced bearing is very laborious and would not be used. In 
practice the calculations would be made with the aid of either 
logarithmic tables or traverse tables. 

The first requirement, whichever method of calculation is adopted, 
is to prepare a sketch of the survey lines with stations lettered or 
numbered and with all bearings and lengths plainly figured along 
the lines to which they refer. The sketch will be all the more useful 



170 



SURVEYING 



if it is roughly plotted to scale with a protractor. It will then show 
the true form and disposition of the lines and will be useful in 
arranging the proper position of the work for plotting. It can also 
be used to check roughly the calculated latitudes and departures, 
and by showing at a glance whether a given line has north or south 
latitude or east or west departure it enables mistakes in sign to be 
avoided. 

The traverse shown on p. 160 has been taken as an example to 
illustrate the calculation of latitudes and departures by the use of 
logarithmic tables and a convenient method of setting down the 
figures and making the calculations is shown in the following table. 

Calculation of Latitudes and Departures. 



Length and 
Bearing. 


Reduced 
Bearing. 


Log Length 

and Log 

Sine. 


Departure. 


Log Length 

and Log 

Cos. 


Latitude. 


AB = 575 
90° 0' 


90° 0' E. 


— 


575 E. 


— 





BC = 660 
46° 0' 


N. 46° r E. 


2-81954 
9-85693 


474-7 E. 


2-81954 
9-84177 






2-67647 


2-66131 


458-5 N. 


CD = 515 
266° 0' 


S.86° 0' W. 


2-71181 

9-99894 

2-71075 


513-7 W. 


2-7U81 

8-84358 






1-55539 


35-9 S. 


DE = 393 
309° 0' 


N. 51° 0' W. 


2-59439 
9-89050 

2-48489 


305-4 W. 


2-59439 
9-79887 






2-39326 


247-3 N. 


EA = 707 
199° 0' 


S. 19° 0' W. 


2-84942 
9-51264 


230-2 W. 


2-84942 
9-97567 






2-36206 


2-82509 


668-5 S. 



PLOTTING BY CO-ORDINATE METHOD. 171 

In the first column the measured values of the length and bearing 
of each survey line are entered. The reduced bearings referred to 
the north and south meridian are then worked out and entered in 
the second column. The third column contains for each survey 
line the log of its length and the log sine of its reduced bearing and 
their sum which is the log of the departure. In the fourth column 
is entered the departure corresponding to the logarithm found in 
the third column. The last two columns contain the figures for 
the calculation of the latitudes. 

It will be noticed that the bearing of AB is 90° or its direction is 
due east. This line, therefore, requires no calculation as its departure 
is just equal to its length and it has no latitude. 

It is essential to rapid and accurate work that the operations 
should be carried out systematically in groups, each group being 
completed for all of the lines before the next is commenced. The 
proper order of operations for the tabulated method of calculation 
shown above is as follows : The reduced bearings are worked out 
for all the lines and the letters designating the quadrants in which 
they occur are written down. The logarithms of all the lengths are 
then looked out from the tables and the value for each survey line 
is inserted both in the log sine column and in the log cosine 
column. Log sines and log cosines of the reduced bearings are then 
abstracted from the tables and inserted under the logs of the 
lengths. Next, all the additions of the logarithms are made, 
thus obtaining the logarithms of the latitudes and departures, and 
from the tables the numbers which correspond to these logarithms 
are found, thus obtaining the latitudes and departures of the 
lines. 

All the figures and calculations of the table should be carefully 
checked before any use is made of the latitudes and departures, as 
it need hardly be emphasised that a mistake in the figures renders 
the results worse than useless. 

Traverse tables give the latitudes and departures of survey fines 
for angles between 0° and 90°, and lengths of 1, 2, 3, up to ten units. 
The values given are simply the natural sines and cosines of the 
angles multiplied by the figures 1, 2, 3, &c. To be of use for 
ordinary surveying purposes they must be given for each minute 
of angle and to the fourth significant figure. To find by means of 
the tables the latitude of a line, say, 473 ft. long write down the 



172 SURVEYING 

tabular values for lengths of 400, 70 and 3 ft. and add them 
together. 

The following is a portion of a traverse table for the angle 
24° 15' :— 



24° 15' 



i 

Lat. Dep. 

0912 0411 



2 
Lat. Dep. 

1-824 0-821 



3 

Lat. Dep. 

2-735 1-232 



Lat. Dep. 

3-647 1-643 



The latitude and departure of a line having a length of 420'3 ft. 
and a reduced bearing of 24° 15' would be found as follows, 
making use of the above line of the tables : — 





Latitude. 


Departure 


400 


364-7 


164-3 


20 


18-24 


8-21 


0-3 


0-27 


0-12 



420-3 383-2 172-6 

For illustration purposes the separate lengths for which the values 
are abstracted from the tables have been shown in the first column 
above. In the actual working out of the latitudes and departures 
these lengths are not written down. 

If a survey line has been measured to the nearest tenth of a foot 
it serves no useful purpose to go beyond the first decimal place in 
working out the latitudes and departures. Thus in the above 
example the latitude and departure for a length of 0-3 ft. as got 
from the table are 0-2735 and 0-1232 respectively, but it is only 
necessary to write down 0-27 and 0-12 in order to obtain the 
desired result. 

Calculation of Co-ordinates. — In order that the co-ordinates of the 
stations may be calculated, one of the stations must be chosen as a 
starting point and its position fixed with reference to the co-ordinate 
axes. That is, suitable values must be fixed for the co-ordinates of 
one of the stations. In the example for which the latitudes and 
departures have been worked out station A has been taken as the 
starting point, its co-ordinates being made 0, 0. The calculations 
for determining the co-ordinates of the other stations are shown in 
the following table : — 



PLOTTING BY CO-ORDINATE METHOD. 

Calculation of Co-ordinates. 



173 





Departure. 


Latitude. 


Corrected 
Departure. 


Corrected 
Latitude. 


_j 


Co-ordinates. 












1 




























E. 


W. 


N. 


S. 


E. 


W. 


N. 


s. 




N. 


E. 




















A 


0-0 


0-0 


AB 


575 


— . 


— . 


— 


574-9 


— . 


— . 


— 


B 


0-0 


574-9 


BO 


474-7 


— 


458-5 


— 


474-6 


— . 


458-1 


— 





458-1 


1049-5 


CD 


— , 


513-7 


— 


35-9 


— 


513-8 


— 


36-0 


D 


4221 


535-7 


DE 


— , 


305-4 


247-3 


— 


— . 


305-5 


247-0 


— 


E 


669-1 


230-2 


EA 


— ■ 


230-2 


— ■ 


668-5 


— 


230-2 


— 


669-1 


A 


0-0 


00 




1049-7 


1049-3 


705-8 


704-4 


1049-5 


1049-5 


705-1 


705-1 










Error 0-4 


Error 1-4 

















In the above table the east and west departures and north and 
south latitudes as calculated are first entered in the separate 
columns shown. Each column is then summed up. If the lengths 
and angles have been measured with perfect accuracy and no 
mistake has been made in the figures the sum of the east departures 
should, for a closed traverse, be equal to the sum of the west depar- 
tures, and likewise the sums of the north and south latitudes should 
balance. In general it will be found that the sum of the east 
departures is not quite equal to the sum of the west departures and 
that there is also a difference between the sums of the latitudes. 
If the difference between the totals is so small as to be inappreciable 
in the length of a plotted survey line the co-ordinates may be 
worked out directly from the figures entered in the first four columns. 
Where, however, the difference is large enough to have a definite 
size if plotted to the scale of the plan the values of the latitudes and 
departures must be adjusted before the co-ordinates are calculated. 
To make clear the distinction between appreciable and inappreciable 
difference suppose that the scale of the plan is 1 in. = 100 ft. A 
surveyed length of 1 ft. will be represented by y^o m - on tne 
paper, and this is about the smallest amount which can be definitely 
plotted. A difference, therefore, of 0-3 ft. in the sums of the lati- 
tudes or departures might be ignored, but a difference of 2-3 ft. 
would be quite appreciable on the plan, and such an error would 



174 SURVEYING 

require to be distributed throughout the survey lines before the 
co-ordinates were calculated. In the example tabulated above the 
east departures exceed the west departures by 0*4 ft. and the north 
latitudes exceed the south latitudes by 1-4 ft. In correcting the 
latitudes and departures it has been assumed that the error will be 
equally distributed throughout the lines. Half the total error in 
departures or 0-2 ft. will, therefore, fall to be distributed amongst 
the east departures and the other half amongst the west departures. 
The correction in this case has been made by subtracting 0-1 ft. 
from the east departures of each of the lines AB and BC and by 
adding 0-1 ft. to the west departures of lines CD and DE. Similarly 
the sum of the north latitudes requires to be reduced by 0-7 ft., and 
the sum of the south latitudes requires to be increased by an equal 
amount. The actual amount of the correction applied to each 
line should be made proportional to its latitude. 

The corrected latitudes and departures are entered in the four 
additional columns as shown. In practice these additional columns 
are usually dispensed with and the corrected latitudes and depar- 
tures are written in red ink above the figures in the latitude and 
departure columns. The co-ordinates of point A are 0, 0. The 
co-ordinates of the other points B, C, D, &c, in the east and west 
direction are arrived at by algebraic summation of the corrected 
departures of the lines AB, BC, CD, &c. The co-ordinates in the 
north and south direction are found in a similar manner. 

In this example the axes have been so chosen that the whole of 
the survey lines lie in the north-east quadrant with respect to the 
origin. This makes the co-ordinates of all the points positive. 
For the sake of simplicity it is in general desirable to arrange that 
the survey points shall all occur in the same quadrant, but not 
necessarily any particular one. 

For purposes of illustration the co-ordinates have been worked 
out in the table on p. 175 for the case in which both axes pass 
through the polygon. Station D has been taken as the starting 
point, its co-ordinates being fixed at E. 200, N. 300. 

The lines and points must as before be taken in consecutive order 
round the polygon from the starting point. Care is required 
in making the subtractions where the departures change from 
E. to W. and where the latitudes change from N. to S. The 
arithmetical work is checked by noting if the co-ordinates of the 



PLOTTING BY CO-ORDINATE METHOD. 



175 





Corrected Departure. 


Corrected Latitude. 




Co-ordinates. 










Station. 


















E. 


w. 


N. 


s. 




B. &W. 


N. &S. 












D 


200-0 E. 


300-0 N. 


DE 





305-5 


247-0 


— 


E 


105-5 W. 


547-0 N. 


EA 





230-2 


— 


669-1 


A 


335-7 W. 


122-1 S. 


AB 


574-9 


— 


— 


— 


B 


239-2 E. 


122-1 S. 


BC 


474-6 


— 


458-1 


— 


C 


713-8 E. 


336-0 N. 


CD 


— 


513-8 


— 


36-0 


D 


200-0 E. I 300-0 N. 



starting point found at the conclusion after working right round the 
polygon agree with the values taken at starting. 



Adjustment of Angular Errors. — It has been assumed in working 
out the preceding example that there is no error in the angular 
work. In observing angles in the ordinary manner with a theodolite 
reading to single minutes an error of some fraction of a minute may 
be expected to occur at each angle. In a series of angles of a 
traverse these small errors will not all tend in the same direction 
so that the total error will not accumulate in direct proportion to 
the number of the sides, but it may nevertheless soon become 
appreciable. Thus in a polygon of a dozen sides in which all the 
interior angles have been read by the direct or separate method it 
need not be expected that the sum of the angles will amount exactly 
to the proper number of right angles. An error of three minutes 
would not indicate careless work and would not necessitate the 
angles being read over again. Before proceeding to calculate 
latitudes and departures, however, the angles would require to be 
adjusted so as to add up correctly, by distributing the error through- 
out the work. 

Where whole circle bearings of the traverse lines have been read 
in the field it will generally be found that a small error has crept in 
round the polygon. If the error is within permissible limits the 
bearings should be adjusted so as to make the final bearing of the 
first fine agree with its original bearing. The method of making the 
adjustment is illustrated in the following table for a set of imaginary 
bearings which show a difference of three minutes between the first 
and final bearings of the commencing fine AB. 



176 



SURVEYING 

Adjustment of Angular Errors. 



Line. 


Forward Bearing. 


Correction. 


Adjusted Bearing. 


1 AB 
IBC 


90° 10' 




90° 10' 


46° 13' 


— 


46° 13' 


(CD 


253° 15' 


— 0° 1' 


253° 14' 


DE 


259° 47' 


-0° 1' 


259° 46' 


(ef 


278° 12' 


— o° r 


278° 11' 


(FG 


221° 35' 


— 0° 2' 


221° 33' 


(GH 


182° 16' 


-0° 2' 


182° 14' 


(HJ 


155° 49' 


-0° 3' 


155° 46' 


JK 


94° 53' 


-0° 3' 


94° 50' 


(ka 


143° 49' 


— 0° 3' 


143° 46' 


Check AB 


90° 13' 


— 


90° 10' 



As the alteration of the bearing of any one line affects the bearings 
of all lines subsequently observed, so the effect of the correction 
of a bearing requires to be carried through all the following bearings. 
The adjustment is, therefore, not quite the same as when the 
separate interior angles are read, as in the latter case only three 
angles would be altered. In adjusting the bearings the assumption 
is made that the error of three minutes has accumulated uniformly 
in working round from side AB to side KA, and the correction is 
made accordingly, but without going to fractions of a minute. 

If a very exact adjustment is required, as might be the case where 
angles have been read to fractions of a minute with an accurate 
theodolite, the correction will not be constant over several lines, 
but will be different for each side. If E is the total angular error 
and N is the number of corrections required (one less than the 
number of sides) then taking the sides in order round the polygon, 
but omitting the first side which remains unaltered, the corrections 

E 2E 

W N 

values may be worked out to seconds of angle, but the correction 
need not be made with any greater degree of precision than has been 
adopted in observing the angles. 

Adjustment of Latitudes and Departures. — The method of adjusting 
latitudes and departures which has been applied in the example 
shown on p. 173 is simple and quite suitable for ordinary purposes. 



will have the following successive values 



— , &c. These 

N 



PLOTTING BY CO-OKDINATE METHOD. 177 

On the assumption, however, of equal chance of error in the 
angular and linear work the proper method of adjustment would be 
to distribute the total error of latitude throughout the lines, making 
each correction proportional to the length of the survey line to 
which it applies, and similarly with the error of departure. The 
corrections are then given by the following equations : — 

Correction in latitude of a side _ Total error in latitude 
Length of side — Total length of traverse 

Correction in departure of a side Total error in departure 

Length of side ~ Total length of traverse 

Weighting the Survey. — Where it is known that certain portions 
of the work have been measured with greater accuracy than others 
the corrections should be judiciously placed at the points where 
the probability of error is greatest. For example, in surveying 
large areas the traverse polygons are commonly arranged with one 
side extending between two stations of a triangulation system. 
The length of this side will have been determined very precisely, 
and hence in adjusting the polygon the corrections should be 
confined to the other sides. 

In a traverse where some lines run over level ground and others 
over steep or rough ground, the probability of error is much greater 
in the latter than in the former, and this should be taken into 
account in adjusting the latitudes and departures. It should also 
be borne in mind that on rough or steep ground the tendency is 
always to measure the lines too long, and hence any adjustment 
made on such lines should preferably be in the direction of making 
them shorter. 

Graphical Method of Correcting for Errors in Chainage only. — In 

many cases, especially in surveys over uneven country, the angular 
work will be much more accurate than the linear work, and in such 
cases it would seem desirable to confine the adjustment to the linear 
work alone, the traverse being made to close by a simple alteration 
of the lengths of the lines. The method is rather laborious. It is 
necessary in the first place to have a fairly accurate diagram of the 
survey lines (such as may be plotted with a protractor), and to have 
the total errors in latitude and departure and the total magnitude of 



178 



SURVEYING 



the closing error worked out. The latter is equal to V E/ + E rf 2 . 
where E^ and E d represent the total errors in latitude and departure 
respectively. The closing error is the same, no matter which point 
of the polygon is taken as the starting point. Fig. 157 shows how 
the closing error might occur in a five-sided polygon, according as 




Fig. 



c '■ d 

157. — Closing Error of a Polygon. 



each angle was in turn made the starting point for the plotting. 
It is evident that for any given polygon there will be at least one 
angle at which the direction of the closing error produced will cut 
the polygon into two roughly equal halves, and in many cases there 
will be two such angles situated on opposite sides of the polygon. 
In Diagram (6) (Fig. 157) the direction of the closing error passes 
roughly through the middle of the polygon. In making the linear 
p. adjustment of the traverse lines it is 

necessary to find the angle at which this 
condition is fulfilled. 

Fig. 158 illustrates the method of 
making the linear correction of the poly- 
gon, the closing error being BB' (shown 
very much exaggerated in the figure) and 
its direction B/ dividing the polygon into 
two portions, forming the separate poly- 
gons BCD/ and B'AE/. The error BB'" is 
bisected in point b. The polygon B'AE/ 
is enlarged to form the similar figure 
baef, the sides ba and ac being parallel to 
The length of each side is increased by 

In the same way the polygon 




Fig. 158. — Linear Adjust 
rnent of Closing Error. 



B'A and AE respectively. 

the fraction ^n~t of its original length. 



B'/ 

BCD/ is reduced to form the similar figure bcdf, each side being 

diminished by the fraction ^f of its original length. The fractional 
reduction in the length of the sides of the polygon BCD/ is for all 



PLOTTING BY CO-ORDINATE METHOD. 



179 



practical purposes the same as the fractional increase in the length 
of the sides of the polygon B'AE/. bcdfea is the complete corrected 
polygon. The bearings of its sides have not been altered, the whole 
adjustment having been effected by slight alteration of the length 
of the sides. 

Fig. 159 illustrates the method of finding by calculation from 
the co-ordinates and by graphical construction the direction and 
magnitude of the closing error and the proper station of the traverse 
at which to deal with it. BCDEA represents the polygon plotted 
with the protractor to a convenient scale. The final side of the 
polygon is simply joined up between the first and last plotted points, 
the closing error being 
meantime ignored. OX 
and OY represent the 
directions of the co- 
ordinate axes. Plot to 
a much enlarged scale 
along OX the distance 
Og equal to the total 
calculated error in de- 
parture. Og will be 
plotted to the right 
or left of the origin, 
according as east or 
west departures are 
the greater. Similarly 
lay off gin to represent 
the total calculated error 




Fig. 159. — Linear Adjustment of Closing Error. 



in latitude. Then Oh represents the 

total closing error in magnitude and direction. Draw a parallel 

to Oh through that station which will cause the polygon to be 

most nearly bisected into equal halves, station B in the figure. 

Scale off the length of B/. Then, for the case shown, the lines 

BA, AE and E/ require to be each increased by the fraction 

half total closing error . . . _• , _ . .. 
wj or their measured length and the lines 

BC, CD, and D/ require each to be diminished by the same fraction 
of their original length. 

The alteration of the length of any survey line will cause a pro- 
portional alteration in its latitude and departure, so that instead 

n2 



180 



SURVEYING 



of dealing with the actual lengths of the lines their latitudes and 
departures, which have been already calculated, may be altered in 
the above proportion. Then, if the operations have been correctly 
performed, the sum of the north latitudes should be equal to the sum 
of the south latitudes, and likewise the sums of the east and west 
departures should balance. 

In the line ED the part E/ requires to be increased and the part 
/D diminished. If point / is near the middle of ED there will, 
therefore, be no alteration necessary in the total length of ED or 




Fig. 160. — Plotting Survey Stations by Co-ordinates. 



in its latitude and departure, 
the line ED will be equal to 



In general, the total correction of 



(D/ - /E) 



half total closing error 



Plotting Survey Stations by Co-ordinates. — The plotting of the 
positions of the survey stations after the co-ordinates have been 
calculated is very simple. Fig. 160 illustrates the plotting of 
the polygon whose co-ordinates have been already worked out on 
p. 173. The easterly co-ordinates, as given in the last column of the 
table, are marked off to scale along the axis OX from the origin 0. 
Oe is the easterly co-ordinate of point E, Od of point D, and so on. 
Perpendiculars are erected at the points e, d, &c, with lengths 
equal to the co-ordinates of the stations, as given in the second last 
column of the table. eE is equal to the north co-ordinate of point E, 



PLOTTING BY CO-OKDINATE METHOD. 



181 






Ml 



dD is equal to the north co-ordinate of point D, and so on. The 
points E, D, &c, so plotted are then joined up by fine pencil lines, 
which form the sides of the traverse. The accuracy of the plotting 
is checked by scaling the lengths of these sides and noting whether 
there is agreement with the measured lengths of the survey lines. 
If no adjustment has been necessary in the latitudes and departures 
the agreement should be almost perfect. If some adjustment has 
been required there will necessarily be a slight discrepancy, which, 
however, should be insignificant. 

The most fruitful source of inaccuracy in plotting by co-ordinates 
lies in the construction of the perpendiculars. Straight-edges 
and set-squares should not be used until they have been tested 
and proved to be 
true. Long perpen- 
diculars should not 
be constructed by 
drawing a portion 
of the length with a 
set-square and then 
extending this line. 
The necessity for 
plotting a large 
number of long per- 
pendiculars can be 
avoided by special 
devices, the best 
method where the stations are numerous being to lay out a 
bounding rectangle with its sides accurately at right angles to 
each other and to sub-divide this into smaller squares or rectangles, 
with sides having a convenient round length. Each station will 
be plotted by small perpendiculars constructed with the set-square 
from the sides of the rectangle in which it occurs. 

A good method of constructing the bounding rectangle and sub- 
dividing it into squares is illustrated in Fig. 161. The base line ef 
is drawn along the middle of the paper. Perpendiculars are erected 
with the set-square near each end of this line, and along these the 
lengths of the sides of the small squares are marked off to scale. 
Lines MN, KL, &c, drawn through these points will be parallel 
to the line ef and at the correct distance apart (for all practical 



Xrf 



Fig. 161. — Construction of Bounding Eectangle. 



182 SURVEYING 

purposes), even although the perpendiculars drawn at the ends of the 
line ef were not absolutely square. To construct an accurate per- 
pendicular to the line ef, two points, a and b, are taken at equal dis- 
tances on each side of the centre of the paper. The distance ab should 
be rather less than the width of the rectangle gh. Using the beam 
compasses or a strip of stout paper and taking a suitable radius, 
strike from the centres a and 6 intersecting arcs to give the points 
c and d near the top and bottom of the sheet. The length of the 
radius should be such as to make the arcs cross each other at about 
90°. The line gh drawn through the points c and d will be an accu- 
rate perpendicular to the line ef. From g and h scale off in both 
directions lengths equal to the sides of the small squares, which 
should not exceed 10 ins., and draw vertical lines through cor- 
responding points on the top and bottom sides. These lines will 
also be perpendicular to the line ef and to the other horizontal 
lines. The accuracy of any square may be tested by measuring 
the lengths of both the diagonals and noting whether they are the 
same. Wherever it can be done without detriment to the plan 
the sides of the squares should be inked in with very fine lines. 
The squares will shrink and expand with the paper, and will form 
a valuable permanent scale over the whole area of the plan. 

If the plan is in the form of a long roll a straight base line should 
be set out along the middle of its length by means of a stretched 
thread. The sides of the squares will be scaled off along this line 
and through points chosen at intervals perpendiculars will be 
constructed as described above, their distance apart being dependent 
on the length of straight edge available. The sub-division into 
squares will then be carried out in the manner described above. 

Alteration of Bearings to suit New Co-ordinate Axes. — It may 

happen that it is not convenient to make one of the co-ordinate 
axes coincide with the direction of the reference meridian. The 
case is illustrated in Fig. 162, where the arrow head represents the 
direction of the meridian to which bearings have been referred, 
and it is desired to plot the area shown on a piece of paper repre- 
sented by the dotted rectangle OXZY. It is evident that for 
plotting purposes the axes should be taken parallel to the directions 
OX and OY. Let the direction of OY be taken as a new meridian. 
Its bearing must be altered to 0° 0', and the bearings of all the 



PLOTTING BY CO-ORDINATE METHOD. 



183 



survey lines must be altered correspondingly. If the bearing of OY 
with respect to the former meridian is a°, then the angle a must be 
subtracted from the whole circle bearings of all the lines to make 
them refer to the new meridian OY. If the bearing of any line is 
less than the angle a add 360° to it before making the subtraction. 
Latitudes and departures worked out from the new bearings will 
be parallel to the lines OX and OY respectively, and the co-ordinates 
can be worked out with respect to these or any parallel lines as 
axes. 



>"0 V 



Connecting Survey Lines from one Sheet to another. — The survey 
of a large area may be readily plotted by the co-ordinate method on 
separate sheets or rolls instead v 

of on one large sheet. The work 
on each sheet is confined with- 
in rectangular boundary lines 
which form common junctions 
between adjacent sheets. Where 
the ends of a survey line occur 
on different sheets a special 
device or calculation is required 
to enable the two portions of 
the line to be plotted. The 
simplest method is to draw on 
a piece of tracing paper a single 
line, corresponding to the 




Fig. 162. — Alteration of Bearings. 



common junction line, and to trace a portion of the reference 
squares on either side of this line. Apply the tracing to one 
of the sheets, making the junction line and sides of the squares 
coincide and transfer the position of the station to the tracing. 
Similarly, apply the tracing in a corresponding position on the 
other sheet and mark' the position of the second station. Draw 
a line connecting the two stations on the tracing paper. Scale 
off its length and note whether it agrees with the measured 
length. If the result is satisfactory apply the tracing again in 
succession to the two sheets and prick through the points where the 
survey line crosses the junction line. The two portions of the 
survey line will be got by joining up these points to the survey 
stations. A further check may now be applied by measuring the 



184 SURVEYING 

separate lengths of these portions and seeing whether they sum 
correctly. The method of finding by calculation the point where 
a survey line crosses the boundary line is explained under " Traverse 
Problems " on p. 214. 

Closing Error and Limits of Error. — The amount of the closing 
error depends in great measure on the consistency with which the 
field work has been carried out. The magnitude of the closing 
error is no criterion of the absolute accuracy of a traverse. For 
example, a polygon may have been measured with a chain of 
incorrect length. If the length of the chain has remained constant 
throughout, its accuracy will not have any effect in causing closing 
error, but the resulting survey will nevertheless be inaccurate. 
Also cases will often occur where the effect of sloping ground may 
be neglected in chaining the survey lines without giving rise to 
closing error as any errors introduced in chaining outwards may 
be balanced by corresponding errors in returning. If, however, 
constant errors, such as result from incorrect length of chain, have 
been as far as possible eliminated and consistent accuracy has been 
aimed at throughout, the closing error will to some extent be an 
indication of the accuracy of the work. 

Using an ordinary good theodolite reading to single minutes the 
error in any single observation should be less than half a minute. 
An error of thirty seconds in the bearing of a line will cause a lateral 
displacement of the extreme station to the extent of about ws&g °f 
the length of the line. This amount of displacement is small com- 
pared with that which is unavoidable in ordinary chaining. On 
undulating ground using the ordinary jointed chain an error of i^oo 
may represent quite good work while with the steel band the error 
should not exceed 2 6oo- On level ground and with special pre- 
cautions much smaller errors than these may be obtained in chaining, 
but generally the angular work will be relatively more accurate 
than the linear. With the assumption of perfect accuracy in the 
bearings the closing error expressed as a fraction of the perimeter of 
the polygon should never be as great as the linear error and should 
hardly exceed half its amount. Thus, if the probable error in 
chaining is tooo the closing error (due to chaining) should not exceed 
2 oo o On undulating ground a total closing error of to^o when the 
chain is used and of 3^00 when the steel band is employed will 



PLOTTING BY CO-ORDINATE METHOD. 185 

represent fairly good work. Closing errors should not exceed half 
these amounts on level ground. Special precautions in the measure- 
ment of lengths and angles are necessary if a closing error of less 
than ^oo is to be consistently attained. 

The limit of error to be set for any particular survey should be 
made to depend on the purpose for which the survey is required and 
on the time and money available for the work. The limit of 
accuracy attainable on paper should also be kept in mind. It is 
very easy to measure a survey line with a steel band with an error 
of less than 50^0- I* is quite impossible to measure by scale a dis- 
tance on a plan to such a degree of accuracy owing to unequal 
contraction and expansion of the paper. 



CHAPTER XIV 

TRIANGULATION 

In this chapter the principles of triangulation by which, starting 
from a measured base line, the positions of a number of points are 
fixed from measurements of angles alone, are considered and the 
application of those principles on a modest scale in practice is also 
dealt with. The important sources of error in reading angles are 
also considered. 

Triangulation. — Starting from the base line AB (Fig. 163), whose 

length has been accurately measured, the position of a point C may 

be fixed by observing the angles at 

C \ £ A and B of the triangle ACB. The 

£} r ■■-"'" / \ \ lengths of the sides AC and CB can 

/ X. \ then be calculated by the principles 

/ \ of trigonometry and these sides may, 

\ / • ., \ therefore, be used as new bases from 

%-* 1 L-^ g which to fix other stations such as 

D and E by angular measurements 

Fig. 163. — Fundamental Prob- • + i_ anmn ^„ TT rpi • j„„ „* .1 „ 

lem in Triangulation. in the Same W- The SldeS ot the 

triangles ADC and BEC may in turn 

be used as base lines from which to fix additional points, and pro- 
ceeding in this way a connected system of triangles may be built 
up the positions of whose corner points are determined by angular 
measurements alone. This method of fixing the positions of a series 
of points over an area to be surveyed is known as triangulation. 

Purpose of Triangulation.— By the triangulation method the 
fixing of an additional station whether it be half a mile or twenty 
miles distant is effected by the observation of a few angles. The 
accuracy of the angular work is practically uninfluenced by the 
distance or the irregularity of the intervening ground, being deter- 
mined almost solely by the precision of the instrument employed 
and the care and precautions bestowed on its use. It need hardly 



TRIANGULATION 187 

be pointed out that the results attainable by triangulation cannot 
be equalled or approached in practice by any method involving 
linear measurements owing to the amount of labour necessary in 
chaining long distances and the unavoidable inaccuracy where the 
ground is uneven. But while triangulation is thus eminently suit- 
able for the fixing of a series of points distributed at considerable 
intervals over a large area it is not adapted to the location of the 
numerous minor stations at close intervals required in the picking 
up of detail. For this purpose some other method such as traverse 
surveying will usually be found suitable. The methods of surveying 
adopted for the location of detail are, however, not in general 
sufficiently reliable when applied to large areas, and hence the pur- 
pose of triangulation becomes evident. That purpose is to locate 
accurately over the area to be surveyed a series of check points 
whose positions so determined are accepted as correct. The minor 
survey lines for the location of the details are then connected as 
directly as possible to these points and any closing error found in 
proceeding from one check point to another is corrected by adjusting 
the intermediate lines and not by any alteration of the positions of 
the triangulation stations. The error in the detail surveying is 
thus prevented from accumulating, being confined to the small 
amount which can occur within any single triangle and this amount 
can easily be kept within reasonable limits. 

For the survey of a country a network of main triangles is set out 
having sides of twenty miles or more in length. The main triangles 
are then divided and sub-divided into smaller triangles the ultimate 
minor triangulation to which the detail surveying is connected 
having sides of an average length of usually less than one mile. 

In the main triangulation instruments of great precision are 
employed, extraordinary precautions are necessary in observing 
the angles and in adjusting their values, and the spherical form of the 
earth must be taken into account. For the minor triangulation 
instruments of ordinary size (5 ins. or 6 ins.) suffice, but greater 
precautions are taken in the reading of angles and the adjustment 
of their values than would be taken in the case of a simple traverse 
survey. The effect of curvature of the earth is inappreciable in 
minor triangulation and may be entirely neglected. 

In the following pages consideration will only be given to the 
subject of minor triangulation on a small scale such as would be 



188 SURVEYING 

suitable for a survey extending over a few square miles, appropriate 
lengths for the sides of the triangles being 2,000 to 5,000 ft. 

Precautions to Ensure Accuracy. — As regards the shape of the 
triangles the equilateral form will give the best results, and the 
practical rule is to have no angle of a triangle less than 30°, which 
necessarily also means that no angle can be greater than 120°. 

The length of one side of a triangle being known the observation 
of the two adjacent angles will enable the lengths of the remaining 
sides to be calculated. It is not necessary to read the third angle 
of the triangle as its value can be found by deducting the sum of the 
other two from 180°, but the reliability and accuracy of the work will 
be enhanced by reading the three angles separately. If their sum 
is found to differ by a slight amount from 180° a correction equal 
to one-third of the difference should be added to or subtracted from 
each angle so as to obtain adjusted values which will add up exactly 
to 180°. 

Many of the stations in a triangulation, such as point C (Fig. 164) 
will form the common apex of several surrounding angles. The 
sum of the angles around any such station is equal to 360°, and 
where any slight discrepancy is found in summing up the observed 
values a suitable correction must be applied. 

Calculation of Lengths of Sides of a Triangle. — The calculation of 
the lengths of the sides is accomplished by the aid of simple trigono- 
metry. In any triangle, such as ABC (Fig. 163), the following 
relationship exists between the lengths of the sides and the sines 
of the angles : — ■ 

a b c 



so that 



sin A sin B sin C 



c sin A , . c sin B 
and b = 



sin C " sin C 

If, therefore, the angles A, B and C, and the length of side c are 
known, the lengths of sides a and b can be calculated from the above 
formulas. Logarithms must be used in making the calculations 
in order to avoid laborious multiplication and division. The 
formulas then become — 

Log a = log c -J- log sin A — log sin C 
Log b = log c -4- log sin B — log sin C 



TRIANGULATION 



189 



It will be noticed that the terms log c — log sin C are common 
to both equations and attention should be paid to this in arranging 
the figures so as to avoid unnecessary calculation. As an example, 
the lengths of the sides a and b will be worked out for a triangle 
having a known side c = 4287-3 ft. long and angles as shown in 
the first column of the following table : — 
The figures are set down in tabular form as follows :— 



Calculation of Sides of Triangles. 



Particulars. 


Logs and Log Sines. 


Logs of Sides. 


Sides. 


c = 4287-3 ft, 
A = 61° 21' 10" 
C = 81° 29' 30" 
B = 37° 09' 20" 


3-63218 
9-99519 


9-943291 
3-63699 { 
9-78102} 


3-58028 
3-41801 


3804-4 ft. = BC 
2618-2 ft. = AC 



In the second column the log of the known side and the log sine 
of the angle opposite to it are inserted. In the third column the 
log sines of the two remaining angles are written down each in its 
proper line. The lower number in the second column (log sin C) 
is then subtracted from the upper number (log c) and the result, 
3-63699, which is equal to log c — log sin C, is written down between 
the numbers in the third column. The two upper numbers in the 
third column are then added together and the result, which is equal 
to log sin A + log c — log sin C and, therefore, equal to log a, is 
placed in the fourth column. The addition of the two lower num- 
bers in column 3 likewise gives the log of side b, and this is also placed 
in the fourth column. The lengths of the sides are obtained by 
looking up from the tables the numbers corresponding to the logs 
of the sides. 

By arranging the work according to the above plan the lengths 
of the sides are arrived at with the least possible amount of calculation 
and figuring. 

For a survey extending over a few square miles it will be quite 
sufficient to employ five-figure logarithms as in the above example. 
For much larger areas logarithms to six or seven places of decimals 
might be used according to the magnitude of the triangulation and 
the degree of accuracy aimed at. 



190 SURVEYING 

Closing Error in Triangulation. — Even when the angles of each 
triangle have been adjusted to sum up to 180° and the angles round 
each station have been corrected so as to sum to 360° there will 
remain room for a certain amount of error in each angle. In a 
minor triangulation the amount of error in an angle may be only 
a few seconds and should not in the most extreme case reach half a 
minute. In consequence of these slight angular errors the calcu- 
lated length of a side at some distance from the base line will vary 
slightly according to the route which has been taken in working 
round to the line. Thus, referring to Fig. 164, the sides of the 
triangles 1, 2, 3 may be successively calculated starting from the 

base line AB and a value for the 

yf , length of the side CF thereby 

y \"---... obtained, thus fixing a position for 

/ \ """---^ station F. But by calculating the 

/ 3 \ 5 /* sides of triangles 4 and 5 a second 

f'--..^ \ yS \ value will be got for the length of 

\ """'---..^ \ yS | side CF. This value will usually 

\ /(K 4 I differ slightly from that previously 

\ 2 / \ I obtained, with the result that the 

\ / \ I position of the station as fixed 

\ / \ I from triangle 5 may be at F' 

\ / \ J instead of at F. The distance 

A * "^ FF' is analogous to the closing 

Fl °- 16 TA n *r4n Err0r " -ror in a traverse polygon. 

It is possible to adjust the angles 
in a set of triangles which form a polygon about a central point so 
that there will be no closing error, but in a small triangulation it 
will not generally be necessary to do this. It will be sufficient to 
take the length of the side as equal to that value which has been 
obtained by the most direct route. 

Field Work. — The principal field operations for a triangulation 
consist of : — 

(a) The establishing and measuring of the base line. 
(6) The fixing of the positions of the stations, 
(c) The reading of the angles with the theodolite. 

Measuring the Base Line. — The base line must be carefully and 
accurately measured, as any error in its length will affect the 



TRIANGULATION 191 

calculated lengths of all the other sides. The degree of accuracy 
to be aimed at will depend on the purpose and magnitude of the 
triangulation and should also be made consistent with the accuracy 
attainable in the angular work. 

In the following description of the measurement of a base line 
the precautions to ensure accuracy are such as have been used for 
a small triangulation where the purpose was simply to obtain 
an accurate plan. 

Smooth level ground is desirable for the location of the base line, 
but a uniform slope is not very objectionable. The length of the 
base line should preferably be about equal to the average length 
of the sides, but a shorter length must of necessity often suffice. 
The base line may be located at any place within the area to be 
surveyed where the conditions are favourable for its accurate 
measurement. It is a good plan to have the base line near one end 
of the survey and to measure also a check base line forming a side 
of a triangle near the other end. 

The measurement will be made with a steel band or steel tape. 
Sufficiently accurate results cannot be obtained with the chain. 
The steel band or tape should before use be carefully compared 
with a reliable standard and its error noted and also the temperature 
at the time of the observation. 

The base line will be ranged out with the theodolite and marked 
at intervals close enough to prevent appreciable deviation from the 
true course in chaining. The ends of the line may be marked by a 
stout wooden peg driven into the ground and having a nail in the 
head to indicate the exact point. If the line is on a slope the vertical 
angle of inclination should be read with the theodolite by sighting 
to a mark at the extreme station fixed at the height of the telescope. 
The chaining will proceed directly along the surface of the ground. 
On soft ground the chain lengths will be marked by sticking in 
arrows and on hard ground, such as a road or footpath, by making 
a fine scratch on the surface and leaving the arrow beside it. Con- 
siderable care must be exercised in planting the arrows and in holding 
the handle up to them, as it is easy to omit the thickness of an arrow 
at each chain length. The band must be kept in good line and 
straight between its ends. A steady and constant pull of 12 or 
15 lbs. should be applied to it at the instant of fixing the arrow. 
The average temperature during the chaining should be observed, 



192 SURVEYING 

and if it is found to be much different from that at which the band 
was tested a correction of the measured length should be made. 

The length of the base line should be chained at least twice, 
once in each direction. If a considerable discrepancy, say, more 
than 1 in 10,000, were found to exist between the two determinations 
the line should be rechained again in each direction. The 
result might be to show that one of the first determinations was 
obviously inaccurate. The mean of the others would then be 
accepted as the most probably accurate result. 

The mean length determined in this manner requires to be cor- 
rected for (a) error in length of band, (6) temperature, and (c) slope 
of ground. 

To illustrate the method of making the corrections, let us 
take the following example : The mean length of a base line as 
chained directly along the surface of the ground was 4368-7 ft. 
The slope of the ground was 1° 14'. The band, when compared 
with a standard at a temperature of 50° F., was found to be 
100-03 ft. long. The base line was chained at a temperature of 
62° F. It is required to find the true length of the base line. 

Each length which has been booked in the result as 100 ft. is in 
reality 100-03 ft. The error due to incorrect length of chain will, 
therefore, be eliminated if the result is multiplied by 1-0003. 

The difference of temperature causes an alteration in the length 
of the band and the correction must, therefore, be applied as for 
incorrect length. A change of 1° F. causes an alteration in the 
length of a piece of steel of about 0-000007 of its length. An 
increase of 12° in temperature will, therefore, cause an expansion 
of the band to the extent of 0-000084 of its previous length, or the 
altered length will be 1-000084 of the original length. The total 
error of the band at time of chaining will, therefore, be 0-0003 
-f 0-000084, or 0-000384 of its length. The true length of the line 
measured along the slope will, therefore, be 4368-7 ft. X 1-000384. 

If a length L be measured along a slope of A°, the true horizontal 
distance X will be equal to L cos A. 

Cos 1° 14' = 0-99977. 

Therefore the true horizontal length of the base line is 4368*7 ft. 
X 1-000384 x 0-99977 =4369-4 ft. 

Broken Base Line. — It may happen that a long enough stretch of 



TRIANGULATION 



193 




suitable ground cannot be found for the accurate measurement 

of a base line in a single straight length. This difficulty may 

sometimes be overcome by arranging the base line in two portions, 

making an angle 

with each other, 

as in Fig. 165. 

The actual 

is then the line „ „ , . 

a n j -j- 1 .lt- Fig. 165. — Broken Base Line. 

AC and its length 

must be calculated. The separate portions AB and BC would be 
chained and the true length of each determined in the manner 
above described. When the theodolite was set up at the points A 
and C for the purpose of ranging out the lines, the opportunity 
would be taken to read the angles BAC and ACB. If BD is per- 
pendicular to AC, then AD = AB cos A and DC = BC cos C, so 
that the whole length AC is equal to AB cos A + BC cos C. 



Enlarging a Base Line. — Where the base line is much smaller than 
the ayerage length of the sides of the triangles it is necessary to 
enlarge it by triangulation. The method is illustrated in Fig. 166. 
AB is the measured base line. Stations C and D are chosen, one on 
either side of AB, and so disposed as to form well-conditioned 
triangles with the points A and B. The angles of the triangles 
having been measured the lengths of the sides may be calculated 
and also the length of the line CD, so that CD may be used as a 

base. CD may 
be about twice 
the length of AB, 
and may be ex- 
panded in a 

C<-'- -IBBBgil - J v "\ n similar manner 

by arranging 
triangles on 
either side of it. 
A The following 

Fig. 166.-EnlargingaBaseLine. example shows 

the method of calculating the length of CD, the particulars of the 
triangles being as entered in the first column of the succeeding table. 
It is only necessary to calculate the length of one side of each triangle 



-.->D 



194 



SURVEYING 



in order to arrive at the length of CD, but it is better to calculate all 
four sides, as this furnishes a check on the accuracy of the results. 



Calculation of Sides. 



Triangle. 


Logs and Log Sines. 


Logs of Sides. 


Triangle ABC. 

AB = 2873-7 ft. 
A = 69° 14' 
C = 47° 17' 
B = 63° 29' 


3-45844 
9-86612 


9-970831 
3-59232 [ 
9-95173} 


3-56315 = log BC 
3-54405 = log AC 


Triangle ABD. 

AB = 2873-7 ft. 
A = 72° 35' 
D = 48° 15' 
B = 59° 10' 


3-45844 

9-87277 


9-97962 1 

3-58567 

9-93382) 


3-56529 = log BD 
3-51949 = log AD 


Calculation of Latitudes and 
Line AB taken as Axis. 


Departures. 


Line and 
Bearing. 


Log Length and 
Log Sine. 


Departure. 


Log Length and 
Log Cosine. 


Latitude. 


AC 

69° 14' 




3-54405 
9-97083 


3272-5 




3-54405 
9-54969 




AD 

72° 35' 


3-51488 

3-51949 
9-97962 


3-09374 

3-51949 
9-47613 


1240-9 




3-49911 
Dep. diff. 


= 


3155-8 


2-99562 
Lat. diff. 


990-0 




3428-3 


= 250-9 


BC 

63° 29' 




3-56315 
9-95173 


3272-5 




3-56315 
9-64978 




BD 

59° 10' 


3-51488 

3-56529 
9-93382 


3-21293 

3-56529 
9-70973 


1632-8 




3-49911 


= 


3155-8 


3-27502 
Lat, diff. 


1883-7 




I 


)ep. diff. 


3428-3 


= 250-9 



TRIANGULATION 195 

The table of latitudes and departures brings out a total departure 
difference between the points C and D of 6428*3 ft. and a total 
latitude difference of 250-9 ft. The length of CD may then be 
calculated from the formula CD 2 = V6428-3 2 + 250-9 2 , which 
gives CD --= 6433-3 ft. 

The second method of finding the length of a line from its latitude 
and departure given under traverse problems (p. 212) is not quite 
so laborious as the above method. The calculation by means of 
logarithms is as follows : 

Log lat. = 2-39950 
Log dep. = 3-80810 



Diff. = 8-59140 = log tan 2° 14' 7" 
Log dep. = 3-80810 

Log cos 2° 14' 7" = 9-99967 



Diff. = 3-80843 = log 6433-3 
The length of the expanded base CD is, therefore, 6433-3 ft. 

Selecting Triangulation Stations. — The facility with which a 
good triangulation system may be arranged depends very much on 
the nature of the country. Open undulating ground presents but 
little difficulty. The opposite is the case in flat wooded territory 
where the erection of special towers may be necessary in order to 
obtain sights over the top of the trees. 

A general examination of the whole area should be made in the 
first place, special attention being paid to those heights and ridges 
which afford a wide view. Any eminence from which an extensive 
view can be obtained all round is sure to block the view in certain 
directions from points at a lower elevation. The higher points 
of the ground are, therefore, the most eminently suited to the pur- 
pose of observation stations and the preliminary examination will 
at once enable certain points to be fixed, which from their com- 
manding position must be used as triangulation stations. Careful 
examination should then be made from those heights with the 
view to fixing probable positions for other stations. These positions 
will in turn be inspected in order to determine the conditions of 
view with respect to the proposed surrounding stations. As it is 

02 



196 SURVEYING 

not easy to judge distances and shapes of triangles accurately by 
the eye an existing map of the area, even if somewhat rough, will 
be of great assistance in enabling a good arrangement of triangles 
to be arrived at with the minimum of trouble. The proposed 
positions of stations having been roughly marked on the map 
and the sides of triangles joined up it can be seen at a glance whether 
the arrangement is good or whether the shape of certain of the 
triangles ought to be improved. 

When the ground has been carefully gone over and provisional 
positions have been fixed for many of the commanding stations 
and a good general idea of the possibilities of the area for the purpose 
of triangulation has been obtained, the laying out of the complete 
triangulation system will be proceeded with. A commencement 
would preferably be made at the base line, the triangulation net- 
work being gradually extended to cover the whole area without 
leaving any gaps or omissions. Where an existing map is available 
the triangles will be laid down on it as they are set out. It will 
sometimes be found that the laying out of a portion of the work in 
the manner judged to be best at the time will give rise to badly 
conditioned triangles further on, which can only be corrected by 
going back and altering some of the work already set out. For 
this reason it is desirable to keep the observation of the angles 
considerably in the rear of the setting out. 

Where no existing map is at hand it will be necessary to plot the 
triangles roughly to a small scale as the setting out proceeds, in 
order that their shape may be seen at a glance. To enable this 
to be done the bearings of the lines may be roughly taken with a 
prismatic compass or with the compass of the theodolite. The 
plotting will be effected by the use of a protractor, starting from 
the base line or from any side whose length has been roughly deter- 
mined. The employment of a small range-finder in conjunction 
with a theodolite or compass will permit of stations being plotted 
roughly from a single bearing and distance. 

For picking out the positions of poles and signals at distances 
of from half a mile to a mile the use of a pair of binoculars is almost 
indispensable. 

A common source of error in the angular work arises when the 
view to a station is obstructed so that the sight must be taken 
to a point on the pole or signal at some height above the ground. If 



TRIANGULATION 197 

the pole is not standing vertical the point observed may not te 
directly above the station, and an error in its position may result. 
It is, therefore, desirable to avoid fixing stations in places where 
tall signals are necessary, and the surveyor may have to decide 
whether in certain cases it would not be expedient to arrange 
somewhat poorer conditioned triangles in order to avoid such 
stations. As the accuracy of a triangulation survey is largely 
dependent on a proper arrangement of the network and on accurate 
sighting to the station points, it will be profitable to expend a 
considerable amount of care and judgment on the proper selection 
and marking of the latter. 

Marking the Stations. — On a small survey the positions of the 
stations will usually be marked by wooden pegs or iron pins, accord- 
ing to the hardness of the ground. The positions of pegs should be 
very carefully referred by accurate measurements to definite 
points or marks on permanent objects, so as to be easily recoverable 
when required. Stone blocks with a mark cut on their upper 
surface, or concrete blocks, in which amiron pin or nail has been set 
may be used as permanent marks. In cultivated ground these 
would be set at such a depth in the ground that their tops would be 
beneath the reach of the plough. 

As temporary sighting marks for the theodolite the ordinary 
6-ft. ranging rods should be used, furnished with a small, bright 
coloured triangular flag at the top to enable their positions to be 
more easily picked out. Distant poles are more rapidly discovered 
by the binoculars than by the theodolite, as the field of view of the 
latter is small. Some poles 10 or 12 ft. long may be useful at 
stations where the view is obstructed near the ground. An assistant 
should be in attendance at such high signals to see that the pole 
is kept vertical while sights are being taken to it. 

An important matter in connection with the reading of the angles 
where the observer at the instrument requires to direct several 
assistants at distances of half a mile and more is the arranging of a 
simple and workable set of communication signals. The out- 
stretched arm and a white handkerchief may serve for simple 
semaphore signals. The number of signals should be confined to 
the minimum that will serve the purpose. It is necessary to have 
a signal to indicate to an assistant when he may lift the pole, 



198 SURVEYING 

another to indicate when he is to come in to the observer at the 
theodolite, and another to indicate when he is to proceed to the 
next station. A source of confusion, annoyance, and delay arises 
when a signal meant for one assistant is taken up by another and 
it is, therefore, very necessary to take such precautions as will 
ensure that a given signal will be accepted only by the person for 
whom it is intended. 

By the exercise of a little foresight and prearrangement a pro- 
gramme of the day's operations may be drawn up which will enable 
definite duties to be assigned to the assistants and definite instruc- 
tions imparted for their guidance, and the necessity for signalling 
may thereby be very much diminished. 

Measurement of Angles. — In triangulation the consistency of the 
work depends entirely on the accuracy of the angular measurements 
and it is, therefore, of importance to take such precautions as will 
ensure that the best use is made of the instrument available and 
the time at the surveyor's disposal. The procedure in measuring 
an angle will usually vary, according as the angle stands by itself or 
occurs as one of a group about a station. The method of " repeti- 
tion," after described, is appropriate to the measurement of a single 
angle, while the method known as " series " is more suitable for the 
measurement of the angles of a group. 

The principal precautions which may be taken in order to attain the 
best results in reading angles with the theodolite are the following : — 

(a) Read both verniers, or if there are three read them all. One 
of the verniers, having a distinguishing mark, will be used as the 
principal index and its reading will be recorded in full. Only the 
minutes and seconds of the other or others need be recorded in order 
to obtain a mean reading. 

(6) Read the angle with the telescope normal and then with it 
inverted and take the mean of the two readings. If the angle is 
repeated several times the same number of observations should be 
made with the telescope normal as with it inverted. 

(c) Read the angle on different portions of the graduated circle 
by setting the vernier index successively to different parts of the 
arc. If there are two verniers the index may conveniently be set 
to zero for the first angle. The reading of both verniers will give 
two determinations of the angle taken on opposite portions of the 



TRI ANGULATION 199 

circle. If the index is then set to 90° and the angle again read on 
both verniers a total of four determinations at equal intervals 
round the circle will have been obtained. If six determinations 
equally spaced are desired the index may be set successivelv to 
zero, 60° and 120°. With a three-vernier theodolite six readings 
equally spaced will be obtained by setting the index to zero and then 
to 60°, or by setting to zero and then to 180°, or by setting succes- 
sively to any two readings which are either 60° or 180° apart. 

(d) Read the angles first clockwise and then anti-clockwise and 
take the mean. 

Precautions (a) and (c) are directed towards eliminating errors 
due to imperfect graduation and centering of the circles. 

Precaution (b) gets rid of errors due to imperfect adjustment 
of the line of collimation and of the horizontal axis of the telescope. 

Precaution (d) tends to eliminate errors due to personal bias in 
observation and to slip of tangent screws. It is not very commonly 
employed in the class of survey in question. 

Repetition Method. — With the telescope normal and starting with 
the index at zero read and record the value of the single angle. 
With the vernier plate clamped at this reading, loosen the lower 
clamp, sight back on the first station, fix the lower clamp and again 
turn the telescope through the angle. The circle reading now gives 
the double of the angle. Repeat the operation a third time with 
telescope normal and then make three more repetitions with the 
telescope inverted. By this process the angle has been added up 
on the circle six times, and one-sixth of the final reading will be 
accepted as a more precise determination than the single reading. 
Of course 360° must be added to the final reading for each complete 
revolution. By taking half the number with the telescope normal 
and the other half with it inverted errors due to incorrect adjustment 
of the line of collimation and of the horizontal axis of the telescope 
are eliminated. 

A development of the above method which tends to still greater 
accuracy in the reading of angles and which may be adopted when 
extreme precision is essential consists in also measuring the exple- 
ment of the angle by means of six repetitions, three with the telescope 
normal and three with it inverted. The angle and its explement 
should together amount to 360°, but there will usually be a small 



200 



SURVEYING 



discrepancy. Apply half the discrepancy as a correction to each 
angle so as to make them sum to 360° and the angles so obtained may 
be accepted as the most probably accurate values. 

Series Method. — The method known as " series," or " reiteration," 
is suitable for the measurement of the angles of a group having a 
common vertex point, and is therefore adapted to most of the work 
in a minor triangulation. 

The method applied to the reading of the angles about the 
point in Fig. 167, assuming that one vernier only was used, 
would be as follows : Set up the instrument at 0, set the index to 
zero and sight the telescope on one of the stations, say, A., Then 
with the lower clamp fixed, loosen the upper clamp, rotate the 
telescope clockwise, sight successively on stations B, C, and D, 
reading the vernier in each case, and finally sight back on A and 
read the vernier. If the final reading differs by more than one 
minute from the starting reading discard the whole set. If the 
closing error is within one minute, assume that it has been intro- 
duced gradually and apply an equal correction cumulatively to 
each angle so as to eliminate the closing error. Make a second set 
of observations starting again with the telescope sighted on station A 
and with the vernier index set this time to 90°. If the telescope was 
normal for the first set invert it for the second set. 

The readings of the various stations and the method of correcting 
the values and obtaining the mean angles are shown in the following 
table : — 

Eeading Angles by Series Method. 





Readings. 










Line. 




Mean 
Reading. 


Correc- 
tion. 


Corrected 
Reading. 


Included 






Angle. 




Normal. 


Inverted. 










OA 


0° 0' 0" 


90° 0' 0" 


0° 0' 0" 


_ 


0° 0' 0") 














74° 36' 11" 


OB 


74° 36' 0" 


164° 36' 0" 


74° 36' 0" 


+ 11" 


74° 36' 11" \ 
162° 41' 7"| 
281° 23' 48" J 














88° 4' 66" 


OC 


162° 40' 30" 


252° 41' 0" 


162° 40' 45" 


+ 22" 














118° 42' 41" 


OD 


281° 23' 0" 


11° 23' 30" 


281° 23' 15" 


+ 33" 














360° 0' 0"J 


78° 36' 12" 


OA 


359° 59' 0" 


89° 59' 30" 


359° 59' 15" 


+ 45" 






Error 45" 


Total 


360° 0' 0" 



TKIANGULATION 201 

As the readings of the second set are booked they should be com- 
pared with the corresponding values of the first set as a check against 
large mistakes. The difference should be 90° in each case to within 
one minute. If the correspondence is satisfactory, make out a 
column of mean readings, taking the average for each line as 
regards the minutes and seconds only and not as regards the 
degrees which are written down as observed in the first set. The 
closing error of the mean readings is found to be 45 seconds, that 
is the final mean reading of the line OA differs by 45 seconds from 
the starting reading. A cumulative correction of 11^ seconds 
should therefore be applied to each reading except the first, that is 
the readings of OB, OC, OD, and the final reading of OA should be 
increased by 11, 22, 33 and 45 seconds 
respectively. The column of corrected 
mean readings is then made out and 
the included angles are obtained by 
successive subtraction. It should be 
noted that while the angles so obtained 
have an appearance of being accurate 
to single seconds and fulfil the condi- 
tion of summing correctly to 360° the 
individual angles may not be accurate 
to within ten seconds or more. 

When a theodolite with two verniers 
reading directly to twenty seconds Fig. 167.— Series Method of 
and by estimation to ten seconds is ea ing ng es * 

employed, two sets of readings are taken as before, one set with 
telescope normal and the other with telescope inverted. Both 
verniers are read so that four values are obtained for the reading 
of each fine. The complete reading is booked for vernier A in each 
case and only the minutes and seconds for vernier B. The mean 
reading of each line is, as regards the minutes and seconds, the 
average of the four observed readings, while the whole degrees are 
as read on vernier A in the first set. The correction of the read- 
ings and calculation of the included angles are made as in the 
example already given above. 

Adjustment of the Angles. — When the angles have all been 
observed and their values obtained in the manners already described, 




202 



SURVEYING 



the condition of affairs will be this : The angles round any station 
will sum to 360° but the three angles of any triangle may not add 
up exactly to 180°. Before calculation of the sides is commenced 
the angles of each triangle must be adjusted so as to sum exactly 
to 180°, without violating the condition as to summation of the 
angles round each station. When the adjustment of the angles 
has been completed the calculation of the sides may be proceeded 
with according to the convenient tabular form shown on p. 189. 

Plotting Triangulation Stations.— When the lengths of the sides 
of the triangles have been computed the stations might be plotted 
on paper by intersecting arcs in the manner described for the 

triangles of a chain survey. 
This method is, however, 
hardly accurate enough for 
an important triangulation, 
and considerable difficulties 
would evidently arise if the 
work required to be distri- 
buted over many separate 
sheets. The co-ordinate 
method is by far the most 
suitable for the plotting of 
the stations of a triangula- 
tion. The first requirement 
is to fix the directions of the 
co-ordinate axes, which will to some extent govern the arrangement 
of the sheets as the edges of these will for convenience be taken 
parallel to the co-ordinate axes. In general it is preferable to adopt 
true north as the direction of one of the axes. The second require- 
ment is to determine the bearing of one of the survey lines, near the 
centre of the area if convenient, with reference to the north and south 
axis. The triangulation may then, for the purpose of calculating 
bearings, be assumed to be divided up into a series of closed traverse 
polygons, arranged to include all the stations. To illustrate this 
Fig. 168 shows a small triangulation, and Fig. 169 shows an arrange- 
ment of polygons suitable for the calculation of bearings and 
co-ordinates. Line AB represents the survey line whose bearing 
has been first determined. When the angles of all the triangles 




Fig. 



168. — Arrangement of Small 
Triangulation. 



TRIANGULATION 



203 



are known the values of the interior angles of the polygon AcdefghB 
can evidently be easily found, and hence the whole circle bearings 
of the sides of the polygon can be determined by the method given 
on p. 147 in connection with traverse surveying, and the reduced 
bearings for purposes of calculating latitudes and departures can 
also be found in the manner already described. In fact, when the 
bearings of the sides have been calculated, the remainder of the work 
of computation and plotting is exactly the same as for a traverse 




Errors in Reading Angles. — The principal sources of error in 
reading angles which arise in connection with the use of the theodolite 
are the following : — 

(a) Theodolite not set 
exactly over the station 
mark. 

(b) Theodolite not cor- 
rectly levelled. 

(c) Telescope not cor- 
rectly focussed. 

(d) Incorrect bisection 
of the station signal 
sighted to. 

(e) Station signal dis- 
placed from its true posi- 
tion, or not vertical. 

(/) Natural causes, including wind, heat, irregular refraction of 
the atmosphere, soft ground, frozen ground, &c. 
The instrument itself may give rise to error in the angular work 
from the following causes : — 

(g) Incorrect adjustment of the line of collimation of 
telescope with respect to its horizontal axis. 

(h) Incorrect adjustment of the horizontal axis of 
telescope. 

(i) Inaccurate graduation and centering of the circle 
verniers. 

Apart from the above sources of error the observer may make 
mistakes, such as the turning of the wrong tangent screw or the 
misreading of a vernier. 



Fig. 169. 



-Polygons for Plotting Small 
Triangulation. 



the 



the 



and 



204 SURVEYING 

Error in Planting the Theodolite. — This error can with little 
trouble be made almost inappreciable in minor triangulation. 
Suppose that the triangulation stations were at distances of 3,400 ft. 
apart (this would not be an inappropriate average length for a 
minor triangulation), then by setting up the theodolite 1 ft. away 
from the true position of a station a maximum error of one minute 
might be introduced in an angle of 60°, and of 1-7 minutes in an 
angle of 120°, and if the theodolite were set up 1 in. away from the 
station the maximum resulting errors in angles of 60° and 120° 
would be five seconds and eight seconds respectively. These latter 
amounts cannot be measured directly with the ordinary 5-in. or 
6-in. theodolite. As it is easy to set up within | in. of the true mark 
without waste of time the error due to incorrect planting should be 
almost inappreciable, and it will evidently serve no useful purpose 
to spend time in adjusting the plumb-bob to the last -^ in. The 
shorter the length of the lines the more accurate does the setting 
require to be. 

Error from Incorrect Levelling. — If the instrument is in adjustment 
and the plate level bubbles are brought as nearly as may be to the 
centre of their runs the error in measuring a horizontal angle due 
to the plates not being exactly level would be almost negligible. 
If the graduated circle is not level and the vertical axis in conse- 
quence not vertical the angle instead of being measured in a hori- 
zontal plane will be measured in an inclined plane, and the result 
will be different. If the points sighted are at the same elevation, 
a slight inclination of the graduated circle will not have much effect 
on the angles read, but if the telescope requires to be tilted up and 
down to sight on high and low points the angles read may be much 
in error. For very important work the final levelling instead of 
being done with the small plate levels should be effected by using 
the large telescope level as this is much more accurate than the 
others. 

Error from Incorrect Focussing. — If the eyepiece is not correctly 
focussed on the cross hairs the line of sight through the telescope 
will be variable according to the position of the eye. If the eye is 
moved from side to side the cross hairs will appear to move relative 
to the object. Under such conditions the angular work is unreliable, 



TRIANGULATION 205 

as the position of the eye may be different for successive sights. 
The procedure to obtain proper focussing has been described 
on p. 127. 



Error from Incorrect Bisection. — If the line of sight, instead of 
striking the centre of a signal, strikes at 1 in. to the side of the centre 
the resulting error will be about five seconds if the distance is 3,400 ft. 
If the signals are ordinary ranging rods, then provided the centre of 
the cross hairs appears on the rod the possible angular error can 
only be two or three seconds. The difficulty with ordinary instru- 
ments when the sights run to 5,000 ft. is that the apparent thickness 
of the hairs becomes comparable with the thickness of the signal, 
and it is, therefore, difficult to determine when the bisection is 
exact, but there is no tendency for the angular error to increase on 
this account when the sights are long. Special precautions should be 
taken to avoid bisection errors when the sights are short. 

Error from Displacement of the Signal. — This may occur if the 
stations are marked by pegs and ranging rods planted beside the 
pegs are used as sighting signals. If the rod is planted vertical and 
on the line from the theodolite through the centre of the peg no error 
will occur, but if the pole is to one side of the correct line an error 
will result. This will most readily happen if a rod which was in 
correct position for a certain sight is inadvertently left in the same 
position for some other sight. 

The greatest possibility of error occurs when a tall pole has to be 
erected as a signal in order that a station may be visible from certain 
others. If a sight is taken to a point near the top of a pole which is 
not vertical the point sighted will not be directly above the true 
station mark, and hence the angle read may not be correct. If only 
a short bit of the top of a pole is visible in the telescope the observer 
may not be able to tell if the pole is vertical, and hence great care 
should be taken to ensure that all signals which must be observed 
at a point above the ground are kept truly vertical during observa- 
tion. If a signal pole requires to be specially tall it should be 
plumbed in two directions with the theodolite and stayed with guy- 
ropes so as to retain the vertical position. 

Errors from Natural Causes. — A high wind may cause the theodo- 
lite to shake and vibrate, and render accurate work impossible. 



206 SURVEYING 

The reading of angles in important work should, therefore, be 
confined to fairly calm weather. 

A strong sun beating on one side of the telescope may cause such 
a difference in temperature between the two sides as to give rise 
to bending of the tube. This would affect the line of sight and, 
if it happened between the readings of two successive lines, an 
error would be introduced into the angle. The precaution may 
be adopted of shading the instrument from the sun, and it will be 
desirable to take the readings of a group as rapidly one after the 
other as possible, so that little time may be allowed for the tem- 
perature state of the instrument to change. 

When the ground has been heated by a strong sun the layers of 
air near the ground are usually of very variable and changing 
temperatures, and objects viewed through such an atmosphere 
appear to waver and wriggle about. A similar effect in a lesser 
degree occurs near the ground at the summit of a ridge or hill 
when a wind is blowing over it. The above effects are due to 
irregular refraction of the atmosphere, and when such conditions 
obtain the reading of angles should not be attempted. 

When the theodolite is set up on soft ground the treading of the 
observer may cause yielding of the ground and disturbance of the 
instrument. The observer should move as little as possible and 
should particularly avoid stepping near the legs of the instrument. 
If the ground is very soft the latter may require to be supported on 
three stakes driven till they are firm. 

If the theodolite is set up on wet ground which has become 
frozen or on ice, the sharp iron-shod points of the legs will gradually 
settle down and the settlement may be irregular. The settlement 
is, however, slow, but, to avoid error in the angles, the observation 
of each group should be carried out as rapidly as possible. 

Error from Incorrect Adjustment of Line of Collimation. — If the 
line of collimation or line of sight of the telescope is not at right 
angles to the horizontal axis it will not revolve in a plane, but will 
describe a flat cone on rotation. This will not give rise to error 
in the angles if the objects sighted are all at one angular elevation, 
but will give rise to very slight error if the objects are at different 
elevations. Error in an angle due to incorrect adjustment of the 
line of collimation will be completely eliminated by taking the mean 






TRIANGULATION 207 

of a number of observations, one-half of which are made with tele- 
scope normal and the other half with telescope inverted. 

Error from Incorrect Adjustment of Horizontal Axis. — If, when, 
the horizontal plates have been accurately levelled, the horizontal 
axis of the telescope is not level the line of sight of the telescope 
will not revolve in a vertical plane, but in an inclined plane, and 
if the observed objects are at different angular elevations the 
angles obtained will be incorrect. This source of error is also 
eliminated by taking observations with telescope normal and with 
telescope inverted, as described in the preceding paragraph. 

Error from Incorrect Graduation, &c. — This should be very small 
in a good instrument. The graduation of a circle may be tested 
by measuring a small angle by repetition until the whole circum- 
ference is traversed, and noting whether any difference is found 
between the successive angles. An occasional difference not 
exceeding the smallest direct reading of the vernier may be ex- 
pected, and the difference between any two readings should not 
exceed this amount. Irregular differences, provided the observa- 
tions were carefully made, would signify that the circle was poorly 
graduated. Graduation errors are partly eliminated by measuring 
the angle on various portions of the circle, as in the " series " 
method. This also partly gets rid of errors due to bad centering of 
the vernier plate with respect to the divided circle. 



CHAPTER XV 

SOME SURVEY, TRAVERSE, AND TRIANGULATION PROBLEMS 

Some examples of the methods of chain surveying applied to the 
solution of special problems were given in Chapter VII. Many of 
these problems can be much more readily and simply solved by 
the use of the theodolite. Some typical problems are dealt with 
in this chapter, mainly for the purpose of educing useful ideas as 
to the methods by which problems of a similar nature may be 
attacked. The problems considered include : setting out a per- 
pendicular from an accessible or inaccessible point to a given line ; 
setting out a line parallel to a given line ; running a straight line 
between two points when an obstacle intervenes ; finding distance 

to an inaccessible point and 

/] distance between two inacces- 

/ i sible points ; traverse prob- 

/ ; lems ; three-point problem ; 

determination of heights by 

the theodolite ; trigonometric 



A 



levelling. 



A D E 



Fig. 170.— Perpendicular from a Point Perpendiculars. — To set out 
to a Line. a p er p en( Ji cu l ar from the given 

point C to the given line AB (Fig. 170). Set up the theodolite at 
a convenient point D on the line AB and measure the angle CDB. 
Next set up at C, sight on D, and lay off an angle DCE equal to 90° 
— CDB. Line CE given by the theodolite will be at right angles 
to AB, since the sum of the angles EDC and DCE is equal to 90°. 

When the given 'point is inaccessible the method shown in Fig. 171 
may be employed. H is the given point, FG is the given line. 
Choose suitable points F and G on the line and measure the angles 
HFG and HGF. Measure also the length FG. Let HK represent 
the perpendicular to the line FG. Calculate first the length of FH . 
In the triangle FHG we have — 



SOME SURVEY PROBLEMS 



209 



Angle FHG = 180° - (HFG + HGF), or H = 180° - (F + G). 



Also 



But 



FH 
sin G 
FK 
FH 



= FG 
sinH 

= cosF 



.-. FH = 



FGsinG. 



sin H 
FK = FH cos F. 



So that FK = 



FGcosFsinG 
sinH 



Point K will then be got 
on the ground by measuring 
off this calculated distance 
along the line from point F. 




-Perpendicular from an 
Inaccessible Point. 



Setting out a Line parallel 
to a given Line. — In Fig. 172, 
AB is the given line, C is the given point. At a suitable point, D, 
on the line AB, measure the angle CDB. Set up the theodolite at C, 
sight on point D, and lay off the angle DCE equal to 180° - CDB. 
The theodolite will then point in a direction parallel to the line 
AB, and the required line CE may be ranged out. 

Running a Line between two Points when an Obstacle Intervenes. — 

It is required to run a straight line between the two points A and B 

(Fig. 173). An obstacle intervenes so that one point cannot be 

seen from the other. 

First method. Choose a point, such as C in the figure, from which 

both points A and B can be 

seen. Measure the distances 

AC and BC. Set up the 

theodolite at C and measure 

the angle ACB. Produce 

the lines AC and BC to E 

and D respectively, making 

CE = CA and CD = CB. 

B Set up the theodolite at E 
172. — Setting out a Parallel Line. „i ,^„„ .-, ■> 

6 and measure the angle 

The triangles DCE and ACB are similar, so that the angle 
CAB is equal to the measured angle CED. The theodolite is, there- 
fore, set up at A and the angle CAA' is laid off equal to CED. Points 
may, therefore, be set in line up to A'. The angle CBA is equal 




Fig 
CED. 



210 



SURVEYING 



to 180° — (CAB + ACB) and this may, therefore, be laid off by the 

theodolite at point B, thus giving the portion of the required line BB'. 

The two portions AA' and BB' so set out will be in one straight line. 

Second method. In Fig. 174, 
F and G are the given points. A 
suitable point H is chosen as 
before and the lines FH and HG 
and the angle at H are measured. 
Then in the triangle FHG two 
sides and the included angle are 
known so that the angles HFG and 
HGF can be calculated by the 
formulas given on p. 354. They 
may then be set off at the points 
F and G respectively so as to give 

the portions of the required line on either side of the obstacle. 
Otherwise let GK be a perpendicular to the line FH produced. 




Fig. 173. — Punning Straight 
Line between two Points. 



Then angle KHG = 6 = 180° - FHG. 

HK = HG cos0. KG = HG sin d. 
KG_ HG sinfl 
KF~~FH + HGcos0* 



Tan HFG =^ = 



The angle HFG corresponding to the above value of the tangent can 

be found from trigonometrical 

tables. The angle HGF can .K 

then be deduced. 

Distance to an Inaccessible 
Point. — It is required to find 
the distance between the 
points A and B (Figs. 175 and 
176), point B being inacces- 
sible. Any convenient base 
line AC may be set out, as in 
Fig. 175. If the length of AC is measured and also the angles 
BAC and ACB, then the length of the side AB can be calculated by 
the method given in the chapter on triangulation. 

When a suitable base line can be set out at right angles to ABb he 
work of calculation is simplified. In Fig. 176, AD is set out at right 




Pio. 



174. — Eunning Straight Line 
between two Points. 



SOME SURVEY PROBLEMS 



211 



angles to AB and its length is measured. The angle ADB is 
measured. 

Then AB = AD tan ADB. 

Distance between two Inaccessible Points. — A and B (Fig. 177) 
represent the two inaccessible points. Measure off a suitable base 




Fig. 176. 
Distance to Inaccessible Point. 

line CD. Measure the angles ACD, BCD, ADC, BDC. Then in 
each of the triangles CAD and CBD one side and the two adjacent 
angles are known so that the remaining sides may be calculated 
by the methods of triangulation. Calculate side CA of the triangle 
CAD, and side CB of the triangle CBD. Then in the triangle ACB 
the two sides CA and CB are known and the included angle ACB is 
also known, so that the 
length of the side AB can be 
calculated by the formula 
appropriate to the case, as 
given on p. 354. 

Distance of a Boat from 
the Shore. — The following 
method of fixing the positions 
of points on the water is used 
in taking soundings. The 
direction of a line of sound- 




Fig. 177. — Distance between Inaccessible 
Points. 



ings is marked by two poles A and D erected on shore (Fig. 178). 
The base line AC is measured off at right angles to AD. The 
boat is rowed out keeping in line with the poles A and D, and 
the positions of soundings are fixed by an observer in the boat 

p2 




212 SURVEYING 

measuring with a sextant the angle made with the points A and C. 
Thus, to fix the position of the boat when at B the observer measures 
the angle ABC. 

Then^=tanABC 5 orAB =ian ^ 

or AB = AC cot ABC. 

Given the Latitude and Departure of a Line to find its Length and 
Bearing.— The graphical solution of this problem is simple. Plot 

the departure ab to scale 
0i (Fig. 179) and lay off the 

latitude be at right angles to 

it. The length of the line 

will be got by scaling ac, and 

its bearing with reference to 

..--''' / the meridian will be obtained 

r h ,---'' / by measuring the angle a at 

I / point c with the protractor. 

The angle a so obtained is the 
i / reduced bearing of the line, 

and the whole circle bearing 

LV' will be got by the converse 

B R of the process shown on 

Fig. 178. — Distance of Boat from Point "" ' . , . n 

on Shore. to the signs of the latitude 

and departure. 

By calculation the length of the line is obtained from the formula 

ac = Vab 2 + be 2 = Vdep. 2 + lat. 2 . 

The angle a is obtained from the formula 

ab departure 
tan a = f- = . *^ 1 • 
be latitude 

When the angle a has been determined by the above method the 
length of the line can be obtained from the formula 

ab departure , . be latitude 

ac = —. — = — *-; and also = = . 

sin a sin a cos a cos a 



SOME SURVEY PROBLEMS 



213 



The calculation of the length by means of this formula is less 
laborious than by the first formula given above. 

Given the Co-ordinates of Two Points to find the Length and Bearing 
of the Line joining them. — Find the latitude and departure of the 
line and use the formulas given in the 
preceding paragraph. The latitude is 
equal to the difference of the two 
meridian co-ordinates treating them 
algebraically, and the departure is simi- 
larly equal to the difference of the two 
co-ordinates in the perpendicular direc- 
tion. 

Example. The co-ordinate distances 
of two points a and c are (N. 537 ; E. 
421) and (K 220 ; W. 209) respectively. 
It is required to find the length and 
bearing of the line ac joining the points. 

Latitude of line = 537 - 

Departure of line = 421 -} 




Fig. 179. — Length of a 
Line from Latitude and 
Departure. 



220 
209 



317 

630 



Length of line = Vdep. 2 + lat. 2 = V630 2 + 317 2 = 705"3 feet. 

T departure 630 1iQfi ., 

Ian a = . r . , — = ^rw = 1 9874. 

latitude 317 _ . . , 

r rom which angle 

a = 63° 17'. The line 
ac is situated with re- 
spect to the co-ordinate 
axes, as shown in Fig. 
180. 

To find the Length and 
Bearing of an Omitted 
Side of a Traverse Poly- 
gon. — The lengths and 
bearings of all the sides 
of a polygon except one 
having been measured 
it is required to calculate 
the length and bearing of the omitted side. The omitted side may 
be considered as a large closing error. Work out the latitudes and 



; 


\ 


3 




N 


^k' 




5^ 


| 


c ^^7--"' 






i'v' 




i<>; 


^i 




i 


<N 






cn; 




j 


! 203 


421 






S 


j 1 



Fig. 180. — Length of Line from Co-ordinates. 



214 



SURVEYING 




..." - -.1 kfzJ*--'* 



Fig. 181. — Eanging a Straight Line. 



departures of all the measured lines. Find the difference between 
the sum of the north latitudes and the sum of the south latitudes. 
This will be the latitude of the omitted side. Similarly find the 

difference between the sums of 
the east and west departures 
to obtain the departure of 
the omitted side. From its 
latitude and departure so found 
calculate the length and bear- 
ing of the omitted side using 
the formulas already given, in 
the manner shown in the 
example which has just been 
worked out. 

This problem may be practically applied in the ranging out of 
a straight line between two points when a large obstruction, such as 
a thick wood, intervenes. A system of traverse lines is run around 
the obstruction to connect the points and the bearings and lengths 
of the lines are measured. Referring to Fig. 181, A and D are the 
points between which the straight line is to be ranged. ABCD is 
the traverse connecting the points. Line AB is taken as the 
meridian direction, A being the origin of co-ordinates. The latitudes 
and departures of the lines 
AB, BC, CD are calculated 
and their algebraic summa- 
tions give the co-ordinates 
of point D and the latitude 
and departure of the line 
AD. From these the angle 
a and the length of the line 
AD can be calculated. The 
angle ft can also be found 
by taking the difference of 
the bearings of the lines CD 
and AD. By setting off the angles a and /3 at the points A and D 
respectively portions of the straight line may be set out and 
extended as the obstruction is cleared away. 

To find where a Survey Line cuts a Parallel to one of the Axes. — 
This problem occurs where a survey is plotted on separate sheets. 





' 


B 






c/ 


! : 


p 






m 






Afy- i£~. 


Ji z 







* x > rt x-x, 


1 j 






- *i 


-— >i 


X 



Fig. 



182. — Survey Line crossing a Sheet 
Boundary. 



SOME SURVEY PROBLEMS 215 

Some of the survey lines will cross the boundary lines of the sheets 
and it is necessary to find the points of crossing. The problem is 
illustrated in Fig. 182. OX and OY are the co-ordinate axes. 
Points A and B are the extremities of the survey line, and their 
co-ordinates are x x y x x 2 y 2 respectively. Line CF represents the 
common joining line of the sheets, parallel to the axis OX and at 
distance Z from it. 

The departure of the line AB = x 2 — x x and its latitude = y 2 — y x . 
The co-ordinates of point C will be found by determining the 
distance AD. 

CD = Z - Ul . 

From similar triangles ^-~ = ^^ 

_ DC X AE _ (Z - y x ) x (x 2 - gj 
^ ~ EB ~ (y 2 - yi ) ' 

The X — co-ordinate of point C is equal to 

Xl + ^ = Xl + «-y^l-*A 

The method is similar when the joining line is parallel to the 
Y-axis, the Y-co-ordinate being then equal to 

yi+ (^^i) ' 

Example. The co-ordinates of the stations at the ends of a 
survey line are 1539-2 E., 268-3 N., and 2275-4 E., 523-7 S. To 
find where the survey line crosses a sheet boundary which is parallel 
to the meridian and at a distance of 2000*0 east of the origin : 

x 2 = 2275-4 y 2 = - 523:7 Z = 2000-0 

x x = 1539-2 y x = 268*3 x t 



''2 



x ± = 736-2 2/2-2/1= -792-0 Z — x 1 = 460-8 



Applying these figures in the formula given above the Y-co- 
ordinate of the point required will be 

268-3 + ^0-8x-792-0 = ^ _ ^ = _^_ 

7oo It 

The minus sign indicates a south co-ordinate, so that the survey 



216 



SURVEYING 



line will cross the joining line of the sheets at a distance of 227-5 

below the X-axis. 

Two Omitted Measurements in a Polygon. — The co-ordinates of a 
polygon can be calculated, provided that not more than two of the 
linear and angular measurements have 
been omitted. The problem has been 
already solved for the case where the 
length and bearing of one side have 
been omitted. We shall now deal with 
the cases where two adjacent sides are 
1 involved. The omitted measurements 
may then be as follows : — 

(a) The lengths of two sides. 
(6) The bearings of two sides, 
(c) The length of one side and the 
bearing of another. 
In Fig. 183 let AC and BC be the 
sides to which the omitted measure- 
ments refer. The general method of procedure in each case is 
to ignore in the first place the sides AC and BC, and to con- 
sider AB as a closing side of the polygon. The latitudes and 
departures of all the known sides of the polygon having been 
calculated, the length and bearing of the omitted side AB are 




Fig. 183.— Two Omitted 
Measurements. 



b/ 



r 




Case (a) Case (b) 

FlG. 184. — Two Omitted Measurements in a Polygon. 



worked out in the manner already explained. Then in the triangle 
ABC, Fig. 184, the known quantities in the several cases are : 

(a) The length of side AB and the angles at A and B. These 
angles require to be deduced from the known bearings of the 
lines AB, AC and BC. 

(6) The lengths of the three sides. 

(c) The lengths of two sides and the angle opposite one of them. 



SOME SURVEY PROBLEMS 217 

This angle also requires to be deduced from the known bearings. 

The three known quantities are sufficient in each case to enable 
the magnitudes of the remaining sides and angles of the triangle 
to be calculated. The necessary trigonometric formulae for the 
solution of the triangles are given on p. 354. 

Case (a). Side c and angles A and B are known. To find lengths 
of sides a and b — 

Find first the angle C, which is equal to 180° — (A + B). 

rn, c sin A , c sin B 

Then a = — = — ~- : o = — - — ~— 

sm C sin C 

Case (6). Sides a, b, c are known. To find angles A and B 

T , a + & + c 

Let s = — —^ — — . 

a 

m, • A Us - b)(s — c) _ . B 1(8 -a) {s -c). 

Then sm -^ = * / v ^ 'and sin -^ = x v — '- 

2 V be 2 V ac 

These formulae enable the angles A and B to be calculated and 
from them the required bearings of the sides AC and BC can be 
deduced. 

Case (c). Sides b and c and angle B are known. To find side a 
and angle A 

Find first the angle C from the formula sin C — — t — • 

Then angle A — 180 — (B + C), and this enables the bearing of 

side AC to be determined. 

&sinA 
Also side a = — = — =5-' 

sm B 

When the lengths and bearings have been calculated by the 
methods above explained, the latitudes and departures of the sides 
to which they refer may be computed and the calculation of the 
co-ordinates may be completed. 

Three-point Problem. — It is sometimes useful to be able to fix 
the position of a point merely by angular observations taken at the 
point. This can be done by reading the angles made with three 
known points suitably situated, as shown in the three diagrams, 
(a), (b) and (c) (Fig. 185). In each case A, B, and C represent 



218 



SURVEYING 



the known points, their relative positions being fixed by the known 
distances d and e and the known 
angle F. is the point to be fixed, 
and to this end the angles D and E 
at point are measured. 

The problem can be solved graphi- 
cally by the use of tracing paper. 
From a point on the tracing paper 
draw three lines of indefinite length, 
making the angles D and E with 
each other so as to represent the 
observed lines OA, OB, and OC. 
Shift the paper about over the 
plotted points A, B, and C, until 
each line passes exactly through its 
point. The required point may 
be then pricked through. 

The problem may also be solved 
by trigonometrical methods, of which 
the simplest consists in calculating 
the angles a and ft 

The four interior angles of the 
quadrilateral ABCO are together 
equal to 360°, so that if S = a + 
ft S = 360° — (D + E + F). 

In triangle OAB we have 

y _ d d sin 



sinD 



ory = 



sin D ' 



and in triangle OBC we have simi- 
e sin /? 



larly y = 



so that 



sin E 
d sin a 



e sin ft 



sin D sin E 
d sin E 



sin a 
Also /3 = S 
sin S cos a ■ 
Substituting for sin /3 in (1) we get, 



(1) 



e sin D 
— a, so that sin /3 = 
- cos S sin a. 



SOME SURVEY PROBLEMS 



219 



or 

which gives 



1 S cos a — cos S sin a 
sin a 

sin S cot a 

cot a = - 



E 



- cos S = 
d sin E 



D 



e sin D' 
d sin E 
e sin D' 

+ cot S . 



(2) 



The value of cot a may be computed from the above formula, 
and the angle a will then be got from a table of cotangents. Angle 
y6 may then be got directly from the formula /3 = S — a. Angle 
/3 may also be found from a formula corresponding to (2), namely : — 
Cot {3 = 

e sin D a 

-j—. — ^—. — q- + COt b. 

d sm E sin S 

When the angles a and 
/3 have been calculated, 
the required point will 
be given by the intersec- 
tion of the lines, got by 
laying off these angles at 
the points A and C respec- 
tively. 

Determination of Heights 
by the Theodolite. — A 

simple case is illustrated 
in Fig. 186. The theodolite is planted at a measured distance 
CD from the vertical object AB, whose height is to be deter- 
mined. A horizontal line of sight CD is given and point D 
is marked. The vertical angle DCB is measured. The height AD 
is measured. Then BD = CD tan 6, and the whole height AB = 
AD + CD tan 6. The height AD instead of being measured 
directly might be found similarly to DB by taking a sight to point A 
and measuring the angle ACD. 

A more general case is shown in Fig. 187. The foot of the object 
cannot be seen from the instrument, and an inclined sight is there- 
fore taken to a point E, and the height EA is measured. Then 
BD = CD tan a. ED = CD tan j3. The whole height AB = BD 
— ED + EA, or AB = CD tan a — CD tan /3 + EA = CD 
(tan a — tan /3) + EA. 




Fig. 186.— Height by Theodolite. 



220 



SURVEYING 



Where the projection of B on the ground is inaccessible, point 
C may be taken as one end of a base line from which point B 

is fixed by triangula- 
^ tion, and the distance 

CD may then be cal- 
culated. 



Trigonometric 

Levelling. — When the 
horizontal distance 
between two points 
is known the relative 
altitudes of the points 
may be determined 
by reading with the 
theodolite the verti- 
cal angle of elevation 




Fig. 187. 



-Height by Theodolite. 

or depression which the one point makes with the other. This method 
of finding altitudes is known as trigonometric levelling. The spheri- 
cal form of the earth must be taken into account, as it appreciably 
affects the results for all distances greater than about one furlong. 

Curvature of the earth. In Fig. 188, ABC represents a line of 
uniform elevation on the 

surface of the earth, is a P___ 

the centre of the earth, 
and AD is a tangent to 
the earth's surface at 
point A, that is, AD corre- 
sponds to the horizontal 
line of sight of a theodo- 
lite set up at A. Let the 
radial fine OB be pro- 
duced to meet line AD in 
point D. BD is vertical 
to the earth's surface at 
point B, and the height 
BD = h is the height 
above the level of point A at which the horizontal line through A 
strikes the vertical through B. Let earth's radius be r, distance 




Curvature of the Earth. 



SOME SURVEY PROBLEMS 



221 



AD be I In triangle OAD, OD 2 = OA 2 + AD 2 , or (r + hf = r 2 + 
I 2 , which gives h = ~ — X"T« 

For distances up to a few miles & is very small compared with 2r 

I 2 
and h may with very little error be taken equal to „-. 

The formula may be put in the form h = C Z 2 , where C is a 
constant depending on the units employed. 

When h is in feet and I is in miles the formula is h = 0-667 I 2 , so 
that when I is equal to one mile h is equal to 0-667 ft. or almost 
exactly 8 ins. 

When h is in feet and I is also in feet the formula becomes 



h = 



2-39 I 2 
10 8 



, or h = 



2-39 P 
100,000,000' 




Fig. 189. — Finding Height of Point above the Theodolite. 

The latter formula would be used when distances are given in 
feet, the former when distances are given in miles. 



To find Difference of Altitude between two Points. — The trigono- 
metric method applied to finding the difference of altitude between 
two given points is illustrated in Fig. 189. It is assumed that the 
points are not much more than one mile apart. A and B are the 
two given points. C is the centre of the telescope axis of the 
theodolite set up at A. EF is a vertical line through point B. 
CD represents a horizontal line of sight at point C, while curving 
line CF is a level line through point C. The theodolite is sighted 
to a point E on a signal erected at B. The vertical angle ECD is 
measured. Then, remembering that point F is at the same level 
as point C we get the total difference of elevation between points 
AandB = AC + FB = AC + FE — EB = AC — EB + FD + DE. 



222 



SURVEYING 



But DE = CD tan a, and FD = 



239 CD 2 
10 8 ' 



239 CD 2 



10 8 



Therefore height of B above A = AC - EB + CD tan a + 

The heights AC and EB must be separately measured. 

Example. Centre of axis of telescope was 4-25 ft. above station 
mark*A. Vertical angle of elevation read to point E was 2° 27' 40" 
and point E was 3-85 ft. above station mark B. To find height of 
B above A, the horizontal distance between the points being 4,381 ft. 



Calculation for ED. 


Calculation for FD. 


Log 4381 = 3-64157 
Log tan 2° 27' 40" = 8-63327 


Log 4381 2 = 7-28314 
Log 2-39 = 0-37840 


Log CD = 2-27484 
CD = 1883 


7-66154 
Log 10 8 = 8-00000 




LogFD =1-66154 
FD =0-4587 



Total height of B above A = 4-25 - 3-85 + 188-30 + 0-46 

= 189-16 ft. 
In the above treatment of this problem it has been assumed that 
the vertical at B is parallel to the vertical at A, and that therefore 



Horizontal line through^ 
K Levei'ljng~~ 




Fig. 190. — Finding Elevation of Point below the Theodolite. 

the angle CDE is a right angle. Also the effect of refraction has 
been ignored. The angle CDE is always greater than a right angle 
(about one minute greater when the distance is one mile) in the case 
of Fig. 189, and less than a right angle in the case of Fig. 190. 



SOME SURVEY PROBLEMS 223 

Refraction is generally allowed for by making a slight reduction of 
the curvature allowance. The error introduced by making the 
foregoing assumptions is negligible when the distance between points 
does not exceed one mile. The errors increase rapidly as the 
distance becomes greater and become important when the distance 
is several miles, but the exact solution applicable to such a case is 
beyond the scope of this book. 

Fig. 190 illustrates the case when the observation is made from 
the higher of the two points. The difference of altitude is then 
equal to BF — AC = BE — AC + ED — DF, 

or = BE — AC+CDtan/3— "^ . 

Note that the allowance for curvature has to be subtracted when, 
as in this case, the observed angle is an angle of depression and added 
when it is an angle of elevation. 



CHAPTER XVI 

LEVELLING 

In this chapter the principles of levelling as practised by the 
civil engineer and surveyor are considered. The principles and use 
of the mechanic's level and the modern water level are dealt with 
briefly. The essential elements of construction of the Dumpy 
level, Wye level and staff in their various forms are explained, 
together with the methods of using them to find the relative eleva- 
tions of two or more points. The methods of entering the readings 
in the level book and computing the levels by the " Rise and Fall " 
and " Instrument Height " methods are also dealt with. 



Levelling. — The operation of levelling has to deal with the 
determination of the relative heights or differences of elevation 
of points or objects, usually at some distance from each other. 
Levelling is used for the two following purposes : — ■ 

(a) To find the relative elevations of existing points. Such, 
for example, as finding the elevations of points on the surface 
of the ground distributed over the site of intended works, informa- 
tion which is usually required to enable the works to be designed. 

(b) To set out points at predetermined differences of elevation. 
This is required in the setting out of all kinds of engineering works. 

Level Surface. — A level surface is accurately defined as a curved ' 
surface which at each point is perpendicular to the direction of 
gravity at that point. A plumb-fine gives the direction of gravity 
at any place. The surface of still water is a truly level surface. A 
horizontal line or a horizontal plane at any point is tangent to a ) 
level surface at that point. The amount of the downward deflection 
of a level surface from a horizontal fine is given by the formulae in 
the preceding chapter, p. 221. 

The amount of the deflection at a distance of one- eighth of a mile 
from the tangent point is \ in. and increases rapidly for greater 



LEVELLING 225 

distances. For most practical purposes, therefore, a horizontal 
plane at a point may be taken as coinciding with a level surface 
through the point, over a circular area having a radius of, say, 
200 yards. 

Levelling Instruments. 

Levelling Instruments. — In the levelling instruments used by the 
surveyor or engineer the action of gravity is employed in various 
ways to indicate a horizontal line. In the Water Level the surface 
of still water in two vertical glass tubes, at a distance apart but 
connected together by a horizontal tube, gives two points at the 
same level. In the Reflecting Level a small glass mirror is sus- 
pended so as to hang with its reflecting surface perfectly vertical. 
A line from an observer's eye placed at some distance from the mirror 
to the centre of the image of the pupil seen in the mirror is then a 
horizontal line. The most common appliance however for indicating 
a horizontal line and the one which is capable of giving by far the 
most accurate results is the Spirit Level. In this the indication 
is given by a bubble of vapour contained in a curved glass tube 
nearly filled with alcohol or ether and sealed at the ends. Under 
the action of gravity the bubble rises to the highest part of the tube 
so that when the bubble has come to rest a tangent to the inner 
surface of the top of the tube at the centre of the bubble is a 
horizontal line. 

Water Level. — An early form of surveyor's levelling instrument 
was the water level. This consisted of two short lengths of glass tube 
connected vertically to the ends of a horizontal metal tube several 
feet long, the whole being mounted on a tripod. The tubes con- 
tained coloured water and, when the instrument was set up and the 
water allowed to come to rest, a horizontal line of sight was obtained 
by bringing the eye into line and level with the two water surfaces 
and looking along them. The instrument was used with a graduated 
staff and sliding vane. 

A modern adaptation of the water level is illustrated in Eig. 191, 
and consists of a pair of glass tubes 2 ft. or so in length fixed in 
graduated metal casing frames having broad bases so that they 
stand erect when placed on the ground. There is a stop-cock and 
hose connection at the foot of each tube so that they can be joined 
together by a length of rubber tubing. The water when at rest will 



226 



SURVEYING 



stand with its surfaces at the same level in the two tubes so that the 
difference of elevation of the two objects on which the tubes stand 
will be got by taking the difference of the readings of the water 
surfaces on the graduated casings. This form of levelling instru- 
ment is very useful for working in confined places, such as narrow 
passages or cellars. The levels in two different cellars, for example, 
can be found provided there is room to pass the rubber tube from 
the one to the other. The instrument is also used by mechanics in 
setting out foundations for machinery, &c. 

Spirit Level.— In the " spirit level," or " level tube," or " bubble 
tube," as it is variously called, the length of the glass tube may 
vary from about 1 in. in the case of a mechanic's hand level to 
7 or 8 ins. in the case of a tube for a sensitive surveyor's level. The 




Fig. 191 



smaller bubble tubes consist simply of a piece of plain round glass 
tube bent to a short radius, and having a single central index mark. 
The appliance indicates " level " when the small bubble is sym- 
metrical about the index mark. The longer and more sensitive 
tubes have their top inner surfaces accurately ground to a flat 
circular curve. The tube is graduated from the centre both ways, 
and the long bubble indicates " level " when its ends are at sym- 
metrical marks on each side of the centre. The bubble in the tube 
of a 14-in. surveyor's level will usually have a length of about 
3 ins., but the length varies considerably with temperature, being 
governed chiefly by the expansion and contraction of the liquid. 
The higher the temperature the greater is the volume of the ex- 
panded liquid, and in consequence the smaller is the length of the 
bubble. 
A line tangent to the inner top surface of the tube, at the centre 



LEVELLING 



227 



of the graduations, which is a horizontal line when the bubble is 
central, is known as the axis of the bubble tube. This is illustrated 
in Fig. 192. The essential requirement of a levelling instrument is 
that the axis of the bubble tube should be parallel to the plane of the 
supports or parallel to the line of sight of the telescope, according 



Axis 




Fig. 192.— Axis of Bubble Tube. 

as the instrument is of the form of the mechanic's level or of the 
surveyor's level. 

Mechanic's Level. — A common form of the mechanic's level is 
illustrated in Fig. 193. The bubble tube is fixed in a hardwood 
casing which is covered over with a metal plate. The plate is open 
over the bubble with the exception of a narrow bar across the centre 
of the tube which serves as an index. There are side slots under 
the covering plate to enable the bubble to be viewed from the 
lateral positions. The extremities of the base are protected by 
metal mountings. In an accurate level the plane of the base must 
be exactly parallel to the axis of the bubble. 

The mechanic's level may be sometimes useful for levelling 
over short distances in confined 
places. The instrument is used, 
in conjunction with a wooden 
straight-edge generally from 6 
to 10 ft. long, as shown in Fig. 
194. Assuming that a point has 
to be established some distance 
off at the same level as the step 
at A, one end of the straight-edge will be placed on the step, the level 
will be laid on the top of the straight-edge at its centre, and the end B 
will be raised or lowered till the bubble is in the centre of its run, 
and then packed up from the ground or in some other way tempo- 
rarily fixed at this level. The top of the support at B will be at 
the same level as the step at A. By shifting the straight-edge 
forward with one end resting on B another support may be fixed 

Q2 




Fig. 193. — Mechanic's Level. 



228 SURVEYING 

ahead at the same level as the starting point, and so on till the whole 
distance is traversed. Errors due to inaccuracy of the level and 
want of parallelism of the straight-edge will be practically elimi- 
nated by reversing the ends of the straight-edge and level at each 
succeeding length. Thus, if A and a be considered as the rear 
ends of the straight-edge and level respectively for the first length 
they should be made the forward ends A' and a' for the second length, 
and so on alternately. By this means any error which may occur 
in an upward direction in one length will be balanced by an equal 
downward error in the next length. 

The mechanic's level is sometimes useful for taking cross-sections 
on very steep ground where the setting up and manipulation of the 
surveyor's level is awkward and troublesome. The method is 
explained on p. 269. 

The Level. — The levelling instrument employed by surveyors 
and engineers for the purpose of determining the relative elevations 



Fig. 194. — Use of Mechanic's Level. 

of points at distances apart is generally known simply as the level. 
Its essential elements are a telescope with a line of sight defined 
by a horizontal cross hair ; a sensitive spirit bubble attached to the 
telescope and arranged with its axis parallel to the line of sight of 
the latter ; a tripod stand to hold up the telescope to a convenient 
height for the observer's eye, and an arrangement of levelling 
screws between the stand and the telescope by which the line of 
sight of the latter can be made horizontal, as shown by the bubble 
of the spirit level coming to the centre of its run. The telescope can 
rotate about a vertical axis so as to command sights in any direction. 
The level is commonly constructed in two different forms, which 
are known respectively as the Dumpy level and the Wye level. A 
variation of the Dumpy level, as manufactured by Messrs. Troughton 
and Simms, may also be noted. An elevation of the ordinary form 
of Dumpy level is shown in Fig. 195, while Figs. 196 and 197 show, 
for the sake of comparison, corresponding elevations of the 
Troughton and Simms Dumpy level and of a Wye level respectively. 



LEVELLING 



229 



Small Cross 
Bubble 



Bubble Tube 



Adjusting 
_/ Screws 



Telescope \ ; 



n p 



Adjusting / 
Sere ws 



Stage 



m W 



&J Diaphragm 

Screws 



Levelling 
Screws 



m 



Parallel Plates 



Diaphragm _ 

.1 rrpw*: ■* r— ^ 



Fig. 195.— 
Dumpy Level. 



Small Cross 




Fig. 196.— 
Troughton and 
f || ^Parallel Simms Pattern 
"zM-Ptztes Leyel _ 



P 



i 



Diaphragm S 



Clip- 



■W 



wy e 



ss 



]ubble Tube 



□ 



Wye 



ii ligg ii ~l 



Clamp 



A ^^- 



^S 



_tf^^=0 K7"5/7ye/7^ Screw 



Fig. 197. — 
Wye Level. 



230 



SURVEYING 



Dumpy Level. — The telescope of the level is generally similar to 
that shown in Fig. 129 for the theodolite, but larger and more 
powerful. Common lengths of telescope are 12, 14, 16, and 18 ins., 




m 



© 



Spiders Webs 

Platinum Wires 

or Lines on Glass. 



Lines on Glass. Platinum Iridium Points. 



Fig. 198. Fig. 199. Fig. 200. 

the 14-in. size being perhaps the most usual for ordinary work. 
The eyepiece is fixed in a tube which usually slides within the tube 
containing the object glass. The movement for focussing the object 
glass generally takes place from the eyepiece end. Fig. 198 illus- 
trates the form of diaphragm having a single cross hair for giving 
the reading on the staff and two vertical hairs to indicate 

whether the staff is held 
erect. Figs. 199 and 200 
show other forms of dia- 
phragm. The diaphragm 
is adjusted in position 
within the inner telescope 
tube so as to have its 
cross hair horizontal when 
the instrument is levelled 
up, and is attached to the 
top and bottom of the tube 
by the two opposing cap- 
stan screws shown near the 
eyepiece end, and its sides 
are held within vertical 
grooves. The capstan 
screws permit of the verti- 




Fig. 



201. — Four-Screw Levelling 
Arrangement. 



cal adjustment of the diaphragm. The bubble tube is supported 
on the top of the telescope, one end of its casing being 
attached by a hinged joint, while the other end is held between a 
pair of capstan nuts on a short length of vertical screw attached 



LEVELLING 



231 



to the top of the telescope tube. This attachment permits of the 
end of the tube being raised or lowered slightly, so as to alter the 
inclination of the bubble axis with respect to the telescope. 

The telescope is supported on a horizontal bar or stage formed, 
preferably, in one piece with a vertical spindle which rotates with- 
in a socket attached to the 
upper parallel plate. 

The parallel plates in the 
four-screw type of levelling 
arrangement are connected 
by a ball-and-socket attach- 
ment, the arrangement being 
as shown in Fig. 201. 

In the three-screw type of 
levelling arrangement, shown 
in Fig. 202, the screws them- 
selves perform the function 
of the ball-and-socket attach- 
ment. The parallel plates 
are replaced by a pair of 
tribrach castings. The thumb 
screws are screwed into the 
extremities of the upper 
casting, and their lower ends 
which have enlargements in 
the form of a ball or a cone 
are held by or clamped to 
the lower casting in such a 
way as to permit of tilting 
to a certain extent, as re- 
quired in the levelling up of 
the instrument. 

The tripod stand is most 
commonly of the solid wooden type, having legs of rounded triangu- 
lar section which fold together into a tapering cylindrical form. 
The framed type of stand is superior and for the sake of extra 
steadiness is sometimes used for the larger instruments, such as 
the 16-in. and the 18-in. 

The following variations of the Dumpy level as above described 




Fig. 



202. — Three-Screw Levelling 
Arrangement. 



232 SURVEYING 

are frequently met with. A small bubble tube is generally fixed 
on top of the telescope in a direction at right angles to the large 
bubble. The preliminary levelling-up may then be done in both 
directions without rotating the telescope, the final precise levelling 
in each direction being afterwards effected by means of the large 
bubble alone. A circular bubble is sometimes fixed for the same 
purpose. 

The large bubble is sometimes suspended beneath the telescope. 
In this position it is not so much exposed to accidental injury as 
when it is on the top of the telescope. 

A small magnetic compass is often fitted on the top of the stage 
underneath the telescope. It is only of very limited use for taking 
bearings, as the telescope of the level cannot be tilted up or down 
to take sights to points which are much above or below the horizontal 
plane through the instrument. 

It is a convenience to have a clamp and tangent screw on the 
vertical axis of the larger power instruments for ease in setting the 
line of sight on to the staff and fixing it there. A clamp and tangent 
screw are almost a necessity where the diaphragm is of the metal- 
point form as this requires very exact setting, and are also indis- 
pensable if the compass is to be used. 

The main requirements of a level in accurate adjustment are the 
following : — 

(a) The line of sight of the telescope should be parallel to the 
bubble axis. 

(b) The bubble axis should be at right angles to the vertical axis 
of rotation. 

Requirement (a) is a necessity to accurate levelling. Require- 
ment (6), however, is for convenience to permit of rapid levelling, 
and accurate results can be obtained although it is not complied 
with. If (b) is fulfilled, then, when the instrument has been set up 
and levelled, the telescope can be rotated so as to point in any direc- 
tion and the bubble will always remain in the centre of its run. A 
series of sights can, therefore, be taken from one position of the 
instrument without further levelling up. If (6) is not fulfilled the 
bubble requires to be brought to the centre by means of the plate 
screws for each new direction of sight. 

The methods of making the adjustments to bring a level into 
compliance with the above-mentioned requirements are described 



LEVELLING 233 

in Chapter XXII. The differences between the several forms of 
level, from the point of view of the person who has to use them, lie 
principally in the provisions made for adjustment. 

Troughton and Simms Level. — In the Troughton and Simms 
pattern of Dumpy level the bubble tube is permanently fixed in 
position on top of the telescope tube by the instrument maker and 
is not intended to be altered or interfered with except in case of 



The telescope is attached to the horizontal stage by a hinge at one 
end and by capstan screws at the other which work against a stiff 
spring. By turning these capstan screws so as to raise or lower the 
end of the telescope the bubble axis may be adjusted so as to be 
perpendicular to the vertical axis of rotation, without affecting the 
relative disposition of the bubble axis to the line of sight. The 
provision for adjustment of the diaphragm is the same as in the 
Dumpy level. 

Wye Level. — In the Wye Level, as illustrated in Fig, 197, the 
telescope is a separate and detachable portion of the instrument. 
Two circular collars of exactly equal diameter are formed on the 
exterior of the telescope tube and these rest in the wyes or supports 
on top of the stage. The wyes are either V-shaped, in which case 
the collars rest on only two points, or (in some modern instruments) 
they are circular. A hinged clip passes over the top of each collar 
to hold the telescope in position and is fastened with a pin or a 
spring catch. The telescope can be lifted out of the wyes and 
replaced end for end, and it can also be placed with the bubble tube 
uppermost as well as underneath, the latter position being the normal 
one and the only one in which the bubble can be read. These 
facilities enable the Wye level to be more easily adjusted than the 
Dumpy. The Y-supports are fastened to the horizontal stage by 
screw attachments which permit of the tilting of the telescope to a 
slight extent as required in adjusting the bubble axis to render it 
perpendicular to the axis of rotation as required in (6). The 
diaphragm of the Wye level is fixed by two pairs of capstan screws 
so as to be adjustable in both the vertical and horizontal directions. 

The Staff. — The simplest form of level staff as regards construction 
and graduation is the solid wooden pattern which is generally made 



234 



SURVEYING 



oc^" 



m\ 



in three lengths with socketed joints giving a total length when put 
together of from 14 to 16 ft. It is divided into feet and tenths of 
a foot by black division lines which are each 
t Jq ft. in thickness. This form of staff is illus- 
trated in Fig. 203. The divisions marking the 
whole foot-lengths are numbered in large figures 
with the exception of the divisions at 5, 10, 
and 15 ft., which, to avoid mistakes, are marked 
by the Roman numerals V, X, and XV respec- 
tively. Each intermediate | ft. is indicated by 
a black diamond on the centre of the graduation 
mark, and sometimes each J ft. is indicated by 
a small black circle. Both sides of the staff 
are graduated alike. The staff is slightly 
hollowed on each face, the cross-section being 
as shown in Fig. 204, with the view to pre- 
venting the rubbing off of the graduations. 

In reading the staff attention must be very 
carefully given to the fact that the telescope 
gives an inverted view. The figures, therefore, 
appear upside down and the graduations run 
from the top of the field of view downwards. 
The operations in reading the staff are : — 

(1) Look for the figure which is apparently 
above the cross hair and note it. 

(2) Count the number of whole tenths of 
a foot down to the division mark imme- 
diately above the cross hair. 

(3) Estimate by the eye the decimal part 
of a space from the latter division mark 
down to the cross hair. 

(4) The separate distances obtained in (1), 
(2) and (3) being added together give the 
staff reading. 
With practice the above operations can be 

done almost simultaneously and the reading 
can be obtained practically at a glance. When 

using the level constantly one becomes scarcely conscious that 

the staff appears upside down. 



KH § 



■= f£ 



tt 



Fig. 203.— Solid 
or Scotch Staff. 




Fig. 204.— Section 
of Staff. 



LEVELLING 



235 



Fig. 205 shows a portion of a staff as seen in the field of view of a 
telescope. The reading shown is 2-73 (not 3-26). The length of 
staff which appears in the field of view is proportional to the distance 
from the level. When the distance is very short the length of staff 
seen may be less than 1 ft., and may so occur that no figure is visible. 
In that case the staff-holder is instructed to slowly lift the staff. 
The observer looking through the telescope at the upper part of the 
field watches the staff as it appears to move downwards and notes 
the figure which first emerges. This gives the number of whole 




Fig. 205. — Portion of Staff seen in Telescope. 

feet, and the staff is then lowered on to the point and the tenths and 
hundredths are read. 

The top of each black division line and not its centre should be 
taken as indicating the exact graduation. The form of staff above 
described is frequently graduated as shown in Fig. 206. Each 
T 2 o ft. is halved by a short division mark, and the exact positions 
of the hundredths are then as shown in the figure. 

The form of staff known as the " Sop with " is illustrated in 
Fig. 207. Usual lengths are from 14 to 18 ft., there being three 
sections which telescope together for convenience in transport. 
It is fully graduated to T J D ft., and elaborately numbered and 



236 



SURVEYING 



: m' 



*&Q. 



Z& 



2:5'.. 



Fig. 206.— Alternative 
Graduation of Staff. 




marked. The graduations are on one side only. The large figures 
shown on the left-hand side in the figure mark the whole foot lengths, 
the top of the figure and top of the horizontal line indicating the 
exact position of the graduation. The smaller figures 1, 3, V, 7, 9, 
which are repeated on every foot length on the 
right-hand side, indicate the odd decimals, the tops 
of the figures again marking the exact graduation, 
and, as the figures are exactly -^ ft. in height, their 
lower edges mark the even decimals. 

As regards the relative advantages of the two 
forms it may be said that 
for very accurate work where 
large instruments are used 
and readings are taken to 
thousandths of a foot it is 
necessary to use a staff fully 
and accurately graduated to 
hundredths. For the great 
bulk of ordinary levelling, 
however, the simple staff 
with open graduations suf- 
fices. It is more easily and 
much more rapidly read, 
especially at the longer sights, 
and gives results which in 
ordinary work are not in- 
ferior in accuracy to those 



| 9 






obtainable with the Sopwith staff. 

Field Work. 
Use of Level and Staff. — In choosing the position l^ 
in which to set up the level the endeavour should 
be made to find a place from which the staff can FlG - . 2 ^-~ ^°P~ 
be read on all the points within working range of 
the instrument whose levels are required. That is, the point on 
which the level is planted should, if possible, be at such an elevation 
that the line of sight of the telescope will not pass above the top of 
the staff nor below the bottom of it when held on any of the points. 
As regards the position in plan, subject to the above, this can be any- 



LEVELLING 237 

where within range, and should be chosen to avoid obstructions 
occurring on the line of sight towards any of the points. 

The instrument should be set up with a fairly wide spread of the 
legs to ensure steadiness. Attention should first be paid to the 
condition of the levelling screws and the focussing of the eyepiece. 
The plates should be made almost parallel by turning the levelling 
screws, and the eyepiece should be focussed on the cross hairs in the 
manner described for the telescope of the theodolite, p. 127. 

Before pressing the legs into the ground bring the instrument 
nearly to the level by moving their points closer in or further out, 
and by swinging them laterally with reference to the tripod head. 
This preliminary levelling will be sufficiently accomplished when the 
whole of the large bubble is visible in any position of the telescope. 
Then press the legs firmly and evenly into the ground one after the 
other, and again bring the bubble nearly central by a slight extra 
pressure on one or two of the legs as required. A little care spent 
on this preliminary levelling by the tripod legs saves considerable 
time in the manipulation of levelling screws, and, what is also of 
importance, saves unnecessary wear and racking of the screws. 
The final and accurate levelling is undertaken by means of the 
levelling screws. The telescope is placed in line over a pair of 
diagonally opposite screws (in the four-screw type), and these are 
turned simultaneously and evenly in opposite directions till the 
bubble is brought to the centre of its run. The telescope is then 
turned through 90°, so as to be in line over the other pair of screws. 
The bubble will have moved away from the centre, and is brought 
back by turning this pair of screws. On turning the telescope back 
into the first position its level condition will be found to have been 
slightly upset by the movement of the second pair of screws. The 
operations of levelling-up in two perpendicular directions must, 
therefore, be repeated a time or two till the bubble is found to 
remain central in every position of the telescope. In making the 
first two levellings by means of the screws it is not advisable to 
spend time in waiting till the bubble comes absolutely to rest and 
exactly to the centre. Bring the instrument as rapidly as possible, 
approximately, to the level in both directions and then proceed to 
exact levelling. 

In the case of the three-screw levelling arrangement the telescope 
is first placed parallel to a pair of screws and then in a perpendicular 



238 SURVEYING 

direction over the remaining screw, the levelling of the bubble 
being accomplished in the former case by turning the two screws 
in opposite directions, and in the latter by turning the single 
screw. 

To find the difference of elevation of two points, A and B, both 
within range of the instrument, say, at distances not exceeding 
300 ft. from the level. The staff is held vertically on point A. 
The observer at the level directs his telescope on to the staff, brings 
it into correct focus by turning the thumb-screw on the right-hand 
side of the tube, and adjusts till the cross hair appears motionless 
against the staff as the eye is moved up and down. Before reading 
the staff, attention should again be paid to the bubble which, 
if not exactly central, should be made so by turning the pair of 
screws which are most nearly in line with the telescope. The staff 
reading is then taken in the manner already described, and booked. 
The point where the cross hair cuts the staff is at the level of the 
line of sight of the telescope, and the staff reading, therefore, gives 
the vertical distance of point A below the horizontal line of sight. 
When the staff reading at A has been taken and checked by repeating 
the reading the staff-holder is signalled to proceed to B. The 
telescope is directed towards and focussed on the staff, the bubble 
is brought to the centre if it is found to have moved, and the staff 
reading is taken all as before. The staff reading on point B gives 
the vertical distance of B below the horizontal line of sight of the 
level. The staff readings on the two points are, therefore, depths 
measured down from the same datum, and the difference of these 
depths will give the difference of elevation of the points. Note that 
the point which has the greater staff reading is at the lower elevation. 

Signals. — The observer at the level can tell by comparison with 
the vertical hairs whether the staff is being held truly vertical or not 
in the plane at right angles to the line of sight. The usual method 
of signalling to the staff -holder that the staff requires plumbing is 
by holding up the arm vertically and slowly inclining it in the 
direction in which the staff should be moved. The observer must 
remember that the directions, as seen in the telescope, are reversed 
so that he must give the signal to tilt the staff further over in the 
direction in which it seems to be already inclined. 

A single wave of the right hand is the customary signal to indicate 



LEVELLING 



239 



that the staff has been read and that the staff- holder is to proceed 
to the next point. 

Both arms held up vertically indicates that the observer purposes 
shifting the level to a new point, and that the staff-holder is to take 
a " change point " as described later. 

Datum. — The most convenient method of stating and comparing 
the elevations of different points is to refer them all to a common 
level surface, known as a " datum surface," or simply " datum." 
The elevation of each point will then be expressed as so many units 
of length, usually feet, above or below the datum. Heights above 
the datum are positive elevations, depths below the datum are 
negative elevations. To avoid the inconvenience of figuring and 
working with negative elevations the datum should always be 
chosen to come below the lowest point whose elevation has to be 
referred to so that all elevations will be positive. 




Fig. 208.— Bench Marks. 

Bench Mark. — A well-defined and permanent object or mark 
whose elevation has been determined so as to be available for future 
use is known as a " bench mark." In connection with the Ordnance 
Survey of Great Britain bench marks known as Ordnance bench 
marks have been established all over the country principally along 
the fines of the highways. They are generally cut near the ground 
on the masonry of permanent buildings, walls, gate pillars, &c, 
and consist of a horizontal V-groove with a broad arrow underneath. 
The centre fine of the horizontal V-groove marks the exact position 
of the determined elevation. A flat plinth is sometimes used as the 
bench mark with a broad arrow only to indicate the exact position 
(Fig. 208). The positions of the Ordnance bench marks and their 
elevations above Ordnance datum are shown on the g^, 2 ^ 00 and 
6 in. to mile Ordnance maps of Britain. The elevations are generally 
reliable, except in mining districts. 

The surveyor in establishing bench marks for his own use will 



240 



SURVEYING 



usually desire to avoid cutting marks. He therefore makes use of 
points such as plinths of buildings, ends of doorsteps where unworn, 
window sills, bases of lamp posts, tops of gate hinges, or any per- 
manent and prominent points which happen convenient to his 
purpose. The exact positions of such bench marks must be care- 
fully recorded by means of sketches and descriptions. 

Ordnance Datum. — All elevations of bench marks, surface of 
ground, contours, &c, shown on the Ordnance Survey maps are 




Fig. 209.— Levels by Eise and Fall Method. 

referred to the " assumed mean level of the sea at Liverpool," 
which is commonly known as " Ordnance datum." It is usual in 
this country to refer the elevations of points, either to Ordnance 
datum or to some datum at a round number of feet above or below 
Ordnance datum. In the case of harbour work the datum Mould 
be taken at, say, 50 or 100 ft. below Ordnance datum to ensure that 
all foundation levels, &c, would come above the datum and, there- 
fore, not require to be expressed as negative elevations. 

Calculation of Levels. — The calculation of " levels," that is, the 
heights of points above a datum, may be made from the staff 



LEVELLING 



241 



readings in two different ways, known respectively as the " rise and 
fall " method and the " instrument height " method, and illustrated 
in Figs. 209 and 210. In Fig. 209 the relative elevations of points A 
and B are fixed by the staff readings of 3-81 ft. and 8-59 ft. respec- 
tively taken with the level in position No. 1, and the relative 
elevations of points B and C are fixed by the staff readings 5-23 and 
3-75 taken from position No. 2. The level of point A with respect 
to a datum is known to be 43-75. The difference of the staff 



A- 43-75 
past. Height - 47- 56 



1-38-97 

5-21 

44-20 




Fig. 210. — Levels by Instrument Height Method. 

readings shows that in proceeding from A to B there is a fall of 
4-78 ft., and in proceeding from B to C there is a rise of 1-48 ft. The 
elevation of point B is, therefore, 4-78 ft. less than the elevation of 
point A and equal to 43-75 — 4-78 or 38-97. The rise from B to C 
added to the elevation just found for B will give the elevation of C, 
which is, therefore, equal to 38-97 -f- 1-48 or 40-45. 

In the " instrument height " method, shown in Fig. 210, the staff 
reading on point A added to the level of point A gives the elevation 
of the line of sight, or the " instrument height," as we call it, for the 
readings taken from position No. 1. The level of any other point 



242 SURVEYING 

sighted from position No. 1 will be got by subtracting its staff reading 
from the instrument height. Instrument height for position No. 1 
= 43-75 + 3-81 = 47-56. Level of point B = 47-56 — 8-59 =38-97. 
The staff reading on B from position 2 added to the level of B will 
give the instrument height for position 2, namely, 38-97 -f 5-23 = 
44-20. The levels of all other points sighted from position 2 will 
be got by subtracting their staff readings from the instrument 
height ; thus, level of point C = 44-20 — 3-75 = 40-45. 

Continuous Levelling. — When two points are at such a distance 
from each other that they cannot both be within range of the level 
at the same time, or when their vertical distance apart is greater 
than the height of the staff, then the difference of elevation of the 
points cannot be determined by a single setting up of the instrument. 
In such cases the distance between the points must be divided into 
stages by intermediate points on which the staff is held, the difference 
of elevation of each succeeding pair of intermediate points being 
found by a separate setting up of the level. This process is known as 
continuous levelling. The term is applicable to any levelling in 
which a connected series of elevations is obtained from a number of 
successive positions of the level. If A and B are two points, say, 
1,600 ft. apart we may choose three intermediate points, Nos. 1, 2 
and 3, dividing the distance into four stages of about 400 ft. each. 
By setting up the level between points A and No. 1 (but not neces- 
sarily in line with them) and reading the staff on each point we may 
determine the difference of elevation of these two points. Similarly, 
by three additional plantings of the instrument we may determine 
the differences of elevation of the pairs of points 1 and 2, 2 and 3, 
3 and B. In proceeding over the several stages from A to B we may 
consider increase of elevation or " rise " as positive, and decrease 
of elevation or " fall " as negative, and the total difference of eleva- 
tion of the end points will then be got by summing algebraically the 
differences of the several stages. In practice, however, the staff 
readings are usually entered in a " level book " specially arranged 
to facilitate the calculation of the levels of the various points by 
the " rise and fall " method or by the " instrument height " method. 

The more general case of continuous levelling where the levels 
of a number of points are, obtained at each setting up of the instru- 
ment in addition to the points required for the carrying forward 



LEVELLING 



243 



of the levels is illustrated in Fig. 211. 
The levelling commences from point 
A, whose elevation above a datum is 
already known, and is continued from 
one position of the instrument to 
another by means of the readings 
taken on the points B, C, D, the staff 
being read on these points in each 
case from two successive positions of 
the level. Such points are termed 
" change points." They may either 
be points whose elevation is wanted, 
or points specially chosen for the pur- 
pose of continuing the levels. Sights 
taken to points such as /, g, h, solely 
for the purpose of finding the eleva- 
tions of these points and not made use 
of in continuing the levels, are known 
as " intermediate sights." " Back- 
sight " and " foresight " are terms 
used to denote the staff readings taken 
on the change points and on the 
commencing and finishing points of 
a series of levels. The commencing 
sight taken from the first position of 
the instrument is a " backsight." 
Of the two sights taken to each change 
point the one which is taken from the 
first position of the instrument is a 
foresight while that which is taken 
from the second position of the instru- 
ment is a backsight. The last 
sight of a series of levels is usually 
reckoned as a foresight. When 
the instrument is set up in a new 
position, in the ordinary course 
the first sight taken will be a 
backsight to the staff which is 
being held on a change point, all 




244 



SURVEYING 



the intermediate sights will be taken next, and the last sight will be 
a foresight on to a new change point. 

For the sake of simplicity of illustration the positions of the 



f 3e 

G/enbrook Water Supply 
Spot Levels For Conduit 






Back 
Sight 


Inter- 
mediate 


Fore 
Sight 


Rise 


Fall 


Reduced 
Levels 


Distance 




4 


42 


















52) 


87 










4_ 


70 













28 


59 


59 










6 


90 










2 


20 


57 


3e 






3 


21 






10 


47 






3 


57 


53 


82 










3 


00 









21 






54 


03 










7 


20 










4 


20 


49 


83 






4. 


77 






II 


ie 






3 


ee 


45 


84 










3 


eo 









87 






46 


71 










6 


30 










2 


40 


44 


31 










2 


+0 






3 


00 






48 


21 






8 


16 






1 


68 





72 






48 


93 










5 


60 






2 


56 






51 


49 










5 


20 









40 






51 


89 














3 


75 


/ 


45 






53 


34 


































zo 


56 






27 


OS 


10 


II 


16 


64 


52 


87 














20 


56 






10 


II 


53 


34 










F< 


ill 


6 


53 


f- 


ill' 


6 


53 


6 


53 
















































































^L 










i 












1 




f 





Reducing Levels. Rise and Fall Method. 

instrument and the positions of the change points have been 
shown as if they occurred in order alternately along a line. In 
levelling work generally the places in which the instrument is set up 
may be disposed in plan in almost any manner with respect to each 



LEVELLING 245 

other, and to the positions of the change points. The terms back- 
sight and foresight must not, therefore, be taken as in any way 
implying direction of sight. 



J.W. ) ^ 
G. B \ 20 14-1 1911 
W. M?L J 




Description 


T.B.M. on 


o/'yoA stone of <ya/e af A 


Po/nrfT 


' F/e/d N°/Zg" .uRuuumFL 


" Cf. 


jyQyffll r 


*s yyyyyyyyyyyA- 


>• h. " " " 


Point 7. 


F/e/aC A*? /3/ 


J 


» A. 





•• I. 


.. 


•< m. 







Point n. 


on access road 


O. 


„ 


Cheek on T.3.M. 53-37 On co a// ox / £. 

















Reducing Levels. Eise and Fall Method. 

Booking and Reducing Levels. 
Booking the Readings and Reducing the Levels. — In most forms of 
level book the left-hand page is arranged for the insertion of the 
staff readings and the calculation of the levels, while the right-hand 



246 SURVEYING 

page is for descriptions of the nature and location of the points and 
for explanatory remarks. The two pages together are reckoned 
and numbered as one page. 

A form of book appropriate to the " rise and fall " method of 
reducing is shown on pages 244 and 245, with the working out of the 
levels from the staff readings shown on Fig. 211. Three columns 
at the left side of the left-hand page are provided for the insertion 
of backsights, intermediate sights, and foresights respectively. 
The readings are entered in the order in which they are taken, 
commencing at the top of the page. The commencing reading, 
4-42, taken on point A, is entered in the backsight column. The 
intermediate sights are then entered on succeeding lines in the column 
provided for them, and the foresight, 10-47, taken to the change 
point B, is entered in the foresight column on the next line below 
the preceding intermediate sight. The next sight taken is a back- 
sight, 3-21, to change point B from position No. 2 of the level, and 
this is inserted in its appropriate column and on the same horizontal 
line as the foresight 10-47. The same procedure is followed in 
entering the set of readings taken from each position of the level. 

The further columns on the left-hand page, headed " Rise," 
" Fall," and " Reduced levels," are used for the working out of 
the levels. The column headed " Distance " is principally used for 
recording the positions of points along a chained line in taking 
longitudinal or cross sections. 

The difference between each staff reading and the next one of 
the same set is calculated and inserted in line with the latter of the 
two readings in the " rise " or " fall " column, according as the 
former reading is greater or less than the latter. Thus the difference 
between the reading on point A and the reading on point / is 0-28 
and is a fall, that is, point / is lower than point A. The number 
0-28 is, therefore, written in the " fall " column in line with the 
reading of point /. Similarly, the rise or fall is worked out and 
entered for each succeeding pair of points. A reference to Fig. 211 
will make clear the proper method of procedure at the change 
points. It is evident that the fall from point g to the change point 
B is the difference between the staff readings on these points taken 
from position No. 1, that is, it is the difference between the last 
intermediate sight of the set and the foresight, while the rise from 
the change point B to the next point h is the difference between 



LEVELLING 247 

the readings taken to these points from position 2, that is, between 
the backsight and the next intermediate sight. 

The figures for " rise " and " fall " worked out thus for all the 
points give the vertical distance of each point above or below the pre- 
ceding one, and if the level of any one point is known the level of the 
next will be obtained by adding its rise or subtracting its fall, as the 
case may be. In the example under consideration, the level of point 
A is known to be 59-87. By subtracting from this the fall of 0-28 to 
point/ we obtain the level of point/. By subtracting the next fall 
of 2-20 from the level of point/ we get the level of point g, and so on. 

When the last staff reading (point E) is entered as a foresight 
the difference between the sum of all the foresights and the sum of 
all the backsights will be the total rise or fall from the first point 
to the last point (A to E). The difference between the sum of all 
the rises and the sum of all the falls entered in the rise and fall 
columns will also give the total rise or fall from the first point to the 
last point. A comparison of the results obtained in these two 
different ways furnishes an efficient check on the calculations of the 
figures in the rise and fall columns as a discrepancy between the 
results can only arise from arithmetical error. This check should 
be applied before the reduced levels are worked out. A complete 
check on the calculation of the reduced levels is also furnished in a 
similar way, for the difference between the level of the first point 
and the level of the last point also gives the total rise or fall between 
these points. The check is complete, because an error in cal- 
culating a reduced level of any point is carried forward as an equal 
error through all the succeeding points, and, therefore, causes an 
error in the last point. These arithmetical checks would only fail 
in the unlikely, but possible, case of two or more errors occurring 
in such a manner as to balance each other. 

A form of level book suitable for the " instrument height " method 
of reducing is shown on p. 248, and, for the sake of comparison, 
the levels have been worked out for the same set of readings as 
before. In this level book, instead of the rise and fall columns 
there is a single column headed " Instrument level," or sometimes 
" Height of instrument " or " Line of collimation." When the 
level is in the position No. 1 its line of sight is 4-42 ft. above point A, 
and the level of the line of sight, or the " instrument height," is, 
therefore, got by adding the backsight 4-42 to the known level 



248 



SURVEYING 



59-87 of point A, giving 64-29. This is entered in its proper column 
in line with the reading of point A. The levels of all the other 
points, read from the position No. 1, are obtained by subtracting the 



37 




ll ' nJ 
1 Hefg 


t 
ht 


Back 
Sight 


Inter- 
mediatt 


Fore 
Sight 


Reduced 
Levels 


Distance 




64 


29 


4 


4Z 








\se 


87 














4 


70 






50 


59 














6 


do 






57 


39 






57 


03 


3 


2/ 






10 


47 


53 


82 














3 


00 






54 


03 














7 


zo 






40 


83 






50 


■6/ 


4 


77 






II 


19 


45 


84 














3 


eo\ 


^ 


7/ 














6 


30\ 




44 


31 














2 


40] 




48 


2/ 






57 


as 


8 


IG 






/ 


68 


48 


93 














5 


GO 






5/ 


40 














5 


ZO 






51 


89 


















3 


75 


53 


34 


































zo 


5G 






27 


03 


59 


87 


















20 


56 


33 


34 














Ft 


7// 


G 


S3 


G 


S3 





































































































Reducing Levele. Instrument Height Method. 

staff readings from this instrument height, the level of the change 
point B being got by subtracting the foresight 10-47. A new instru- 
ment height must now be found for position No. 2, and a reference 
to Fig. 211 will make it plain that this will be got by adding the 



LEVELLING 249 

backsight 3-21 on point B to the level 53*82, just obtained for this 
point. The rule for the procedure at the change points is, therefore, 
subtract the foresight from the instrument height to get the reduced 
level of the change point, then add the backsight to this reduced 
level to obtain the new instrument height applicable to the next set 
of intermediate sights and the next foresight. 

All the instrument heights and levels of the change points may be 
worked out and the level of the last point obtained before any of 
the levels are reduced from the intermediate sights, and this pro- 
cedure is preferable, as it permits of a check being applied to so 
much of the calculations. If the difference between the levels 
of the first and last points is found to be equal to the difference 
between the sum of the foresights and the sum of the backsights, 
then the calculation of the instrument heights and change point 
levels may be accepted as correct and the reduction of the levels of 
the intermediate points may be proceeded with. It is evident that 
the calculation for each of the latter points is quite independent 
and does not affect the calculated level of any other point, so that 
the instrument height method of reducing does not in itself furnish 
any check on the accuracy of the calculated levels of the inter- 
mediate points. It is, therefore, advisable that the levels worked 
out by this method should be checked by a second person to ensure 
their accuracy, and the best way of effecting this is to have the 
reduced levels worked out independently on a strip of paper, placed 
so as to cover up the first set of figures, the two sets of results being 
afterwards compared and any discrepancies investigated. 

Another method of checking the levels of intermediate points 
is as follows : If there are n intermediate levels in the set between 
backsight and foresight, subtract the sum of the intermediate staff 
readings from n times the instrument height. The result, if the 
reducing is correct, will be equal to the sum of the reduced levels 
of the intermediate points. 

Sometimes, instead of having three columns for the staff readings, 
only two are used, the first for the backsights and the second for the 
intermediate and foresights, the last reading of each group in the 
second column being the foresight. 

The staff readings may also be booked in a single column, pro- 
vided some method is adopted for clearly and unmistakably dis- 
tinguishing between foresights and backsights. 



250 SURVEYING 

One method is to draw a line underneath each foresight, that is, 
beneath each final reading taken on a change point before shifting 
the level. The following reading placed beneath the line will then be 
the backsight. Another method is to mark each foresight with an 
X placed alongside. The accidental omission of any distinction 
between foresight and backsight would, of course, be a serious 
blunder, occasioning wrong results in the reduced levels. For this 
reason the single-column method of booking is unsatisfactory, but 
where it is used the additional precaution should be adopted of 
carefully noting the change points on the description page. 

The three-column method of booking is the usual one, and is the 
most generally satisfactory. 

Whichever method of booking is used the reduction of the levels 
may be made by either the " rise and fall " method or the " instru- 
ment height " method, in the maimers already described. 

As regards the comparative advantages of the " rise and fall " 
and " instrument height " methods of reducing, the latter is the more 
direct and rapid method as it involves very much less arithmetical 
work. The " rise and fall " method, on the other hand, possesses 
the very great advantage of easily furnishing an almost perfect check 
on the whole of the arithmetical work, whereas the other method 
provides in itself a perfect arithmetical check for the levels of the 
change points only. The two methods indicated above for 
separately checking intermediate levels which have been reduced 
by the " instrument height ' ' method are somewhat cumbersome. 
The effect of a mistake in the arithmetic is worth noting in the 
two cases. In the " rise and fall " method, if one is careful to 
apply the checks, the occurrence of the mistake will be detected 
at the foot of the page on which it occurs. The calculations must 
then be gone over again, starting at the top of the page until the 
point is found at which the mistake occurs. The reduced levels 
from this point downwards must then be rubbed out and a fresh 
set of values calculated and inserted, and the result checked at the 
foot of the page. In the case of the " instrument height " method 
an error in the reduced level calculated from an intermediate sight 
can, as already explained, only be detected by the application of 
an independent check, but, when found, it merely involves the 
rubbing out and correction of the particular reduced level involved. 



CHAPTER XVII 

ERRORS IN LEVELLING 

In this chapter consideration is given to the various errors likely 
to arise in levelling operations, as enumerated in the following list, 
and to the relative importance of these errors. The subject is of 
very great importance and is deserving of most careful study. The 
surveyor who has a true appreciation of the various circumstances 
which conduce to the occurrence of error and of the relative impor- 
tance of the resulting errors will be best able to arrange his work so 
as to obtain the best results from given expenditure of time and 



Errors in Levelling. — The principal sources of error in levelling 
are the following : — 

(a) Faulty adjustment of the level. 

(6) Mistakes and carelessness in use of level. 

(c) Errors resulting from the staff and its manipulation. 

(d) Inaccuracies and mistakes in reading the staff and mistakes 
in booking the readings. 

(e) Curvature and refraction and other natural sources. 
(/) Mistakes in reducing the levels. 

Faulty Adjustment of the Level. — If the level is not in correct 
adjustment the line of sight will not be horizontal when the bubble 
is central. The error caused thereby in any staff reading will be 
proportional to the distance of the staff from the instrument. The 
error will be equal in amount for all positions which are at equal 
distances from the instrument, or in other words the line of sight 
will strike the staff at points of equal elevation if the distances are 
the same. The effect of imperfect adjustment will, therefore, not 
be cumulative if the backsight and foresight for each position of the 
instrument are made practically equal in length, but there will be 
a certain amount of error in the various intermediate sights if these 



252 SURVEYING 

are of unequal lengths. If the backsights happen to be consistently 
of greater length than the foresights or vice versa, as tends to be the 
case in levelling continuously uphill or downhill, the error due to 
imperfect adjustment will be cumulative. 

An annoying source of error occurs when the inner tube of the 
telescope does not move exactly parallel to the outer tube, as shown 
by the cross hair appearing to move up or down the staff as the 
thumb-screw is turned for focussing the object glass. This causes 
an error in the staff readings which varies with the length of sight. 
Error will be largely eliminated if the sights can be arranged of 
nearly equal length so that one setting of the focus suffices for each 
position of the level. 

Mistakes and Carelessness in Use of Level. — The bubble may not be 
brought exactly to the centre of its run in levelling up the instru- 
ment. This may happen if the bubble is a sluggish one and is read 
before it has come quite to rest, or if it is viewed from a slanting 
direction and not from a position exactly square to the direction of 
the telescope. For important sights the bubble should be carefully 
examined after the staff is read as well as before it. 

If the eyepiece is not correctly focussed on the cross hairs an error 
of variable amount may result in the staff readings. The value of 
the reading will depend on the position of the eye at the moment. 

Error may also arise due to the shifting or settlement of the level 
if it has not been planted firmly enough in the ground, or if it should 
be accidentally touched or jarred, or if the observer should tread 
too near the legs. 

The lifting of the level before the foresight has been taken is a 
serious blunder. 

Errors Resulting from the Staff and its Manipulation. — The error 
of graduation of a staff should be almost negligible, but it is never- 
theless advisable to test the staff occasionally with a steel tape. 
Alteration of length may occur due to wear at the joints, or, in the 
socketed pattern, due to the lengths not being driven home, or due 
to dirt getting into the sockets. A small error in the length of the 
staff will cause an accumulative error in the results if the levelling 
is continuously uphill or downhill. An error of \ in. in 10 ft. would 
cause an error of fully 6 ins. in levelling through a vertical height 
of 500 ft. 



ERRORS IN LEVELLING 



253 



The error due to alteration in length of a well-seasoned wooden 
staff with change of temperature is for most practical purposes 
negligible. 

Error will be caused if a layer of dirt is allowed at times to adhere 
to the bottom of the staff, but a layer of constant thickness per- 
manently adhering to the bottom would cause no error. 

The principal source of error in connection with the use of the 
staff arises from its not being held truly vertical. In Fig. 212 the 
correct distance of the point C below the horizontal line of sight of 
the level is the length CA read off on the staff held vertically on 
point C. For an inclined position, such as CB, the reading obtained 
will be too great by an amount represented by the length BA\ 
The amount of the error will evidently become increasingly great 




Fig. 212.— Staff not held Vertical. 



as the inclination of the staff to the vertical becomes greater, while 
for a given inclination of staff the error will be in direct proportion 
to the height of the reading on the staff. 

The correct reading of the staff will be obtained by waving it on 
both sides of the vertical about point C towards and from the level, 
and noting the lowest reading. This precaution should be adopted 
for all important points and especially for change points. 

It is not desirable, however, to sway the staff for readings near 
the ground, say, within the lower 2 ft. of the staff, as the graduated 
face may be raised as shown in Fig. 213, thereby giving a reading 
A'B' which is smaller than the reading AB obtained when the 
staff is vertical. This effect occurs most pronouncedly with a staff 
having a comparatively broad base, such as the Sopwith. 

Error due to inclination of the staff is practically always in the 



254 



SURVEYING 



direction of making the reading too great. The general effect will, 
therefore, be the same as if the readings were taken with a shrunk 
staff. In proceeding continuously uphill or downhill, the back- 
sights will be constantly greater than the foresights or vice versd, so 
that if any error due to inclination occurs it will tend to be cumu- 
lative. In a stretch of levelling where there is no great change of 
altitude errors due to inclination will tend to be compensative. 

Slight settlement of the staff downwards occurring at the change 
points between the reading of the foresight and the backsight will 
cause an error equal to the amount of the settlement. The error 
will be carried forward, and recurring errors due to this cause will 

be in the same direction and 
will be cumulative. The staff 
should always be held on firm 
/ and definite change points and 

should preferably never be 
removed between the reading 
of the foresight and the back- 
sight. 



Graduated 
Face 



line Qp^ Sight 



■'A' 



Fig. 213.— Error at Bottom of Staff. 



Inaccuracies and Mistakes in 
Reading the Staff and Mistakes 
in Booking the Readings. — Re- 
ferring to Fig. 205, which shows 
a portion of a staff as seen in- 
verted in the telescope of a level, the correct reading of the centre 
hair is 2*73. A common mistake with beginners is to read upwards 
instead of downwards, thus obtaining the wrong reading 326. 
The mistake may also readily be made of reading the decimal 
portion correctly but taking the wrong whole numeral. For 
example, in taking the reading shown in the figure, the numeral 3 
being nearest the centre cross hair is most prominent to the eye and 
is apt to be set down instead of the correct numeral 2. The 
incorrect reading obtained would thus be 3-73. 

If the telescope is furnished with stadia hairs, the mistake may be 

made of reading the wrong cross hair. The upper hair (as seen in 

the telescope) is the one which is most liable to be read, and in the 

case illustrated the wrong reading 2-11 would result. 

The observer requires to be alive to a fruitful source of error when 



ERRORS IN LEVELLING 255 

the cross hair occurs in the first tenth of any foot-length. The 
error consists in estimating the decimal fraction of the first tenth 
and setting it down in the book as the decimal fraction of a whole 
foot. Thus if the true reading were 2-04 ft. it might be booked 
incorrectly as 2-40 ft. 

The longer the sight the greater is the apparent thickness of the 
cross hair compared with the graduations of the staff so that at 
long sights the true reading may be indefinite to a few hundredths 
of a foot, and a small amount of error may result in consequence. 
Long sights should not be taken to any points whose level requires 
to be accurately determined. 

Mistakes in reading the staff can only be avoided by the exercise 
of constant care. The likely sources of error as noted above 
should be kept in mind with a view to their avoidance, and in the 
case of important sights the reading should be repeated once or twice 

\B A C /Horizontal Line 



Fig. 214. — Effect of Curvature and Eefraction. 

after it has been booked, the eye being removed from the telescope 
between the separate readings. 

A mistake in booking which is liable to happen with beginners 
is the setting down of the foresight in the backsight column or vice 
versd. The mistake of omitting to book a reading altogether may 
occur if the observer is spoken to or disturbed after he has read the 
staff but before he has entered the result in the notebook. The 
mistake is sometimes made of unconsciously recording numbers 
with a pair of figures interchanged — for example, writing down 7*53 
when the figure in the mind is 7*35. 

The stallholder may hold the staff on a wrong point, or may take 
a group of points in wrong rotation. The result in each case will 
be equivalent to an error of booking. 

Curvature, Refraction, and other Natural Sources. — Fig. 214 illus- 
trates the effects of curvature and refraction. The actual line of 



256 SURVEYING 

sight is represented by the dotted line DAD' which curves slightly 
downwards from a true horizontal line BAC. The curved line EAE' 
represents a level line parallel to the surface of the earth. 

The deflection of the actual line of sight below the horizontal at 
any point is about one-seventh of the deflection of a level line at 
the same point, but the effect of refraction varies slightly with the 
state of the atmosphere. The combined effect of curvature and 
refraction is expressed sufficiently closely by the following 
formula : — 

F2 



C = 



48,000,000 



where F is the length of sight in feet, and C is also given in feet. 
Values of the correction C for various distances F are given below. 
Fin feet = 200 400 600 800 1,000 2,000 3,000 4,000 
C in feet = -001 -003 -007 -013 -021 -083 -187 -333 
From the above table it is seen that the error due to neglecting 
the effect of curvature and refraction is very small for all ordinary 
lengths of sight. For points at equal distances from the level, such 
as F and G in the diagram, the errors will be the same in amount, 
so that if equal backsights and foresights are taken no accumulation 
of error will result. If the foresights are taken consistently longer 
than the backsights or vice versa, as may happen in levelling con- 
tinuously downhill or uphill, the error due to curvature and refrac- 
tion will be cumulative. If, for example, all the foresights are 
taken 100 ft. long and all the backsights are taken 400 ft. long the 
resulting error from this cause will amount to -02 or -03 ft. in a mile 
of levelling. 

If the sights are limited to moderate lengths, and the backsight 
and foresight for each setting of the level are made approximately 
of equal length, as judged by the eye or measured by pacing, the 
error due to curvature and refraction will, for most practical 
purposes, be negligible. 

Other natural sources which may give rise to error in levelling are 
high wind, sun producing atmosphere of variable temperature, 
sun causing variable heating of the level, frost, and thaw. High 
wind may cause shaking of the instrument to such an extent as to 
render accurate levelling impossible. The remedy is either to 
shelter the level, or wait till the wind falls. 



ERRORS IN LEVELLING 257 

When the ground is being heated by a strong sun, and causing 
in turn the warming of the adjacent layers of the atmosphere, the 
latter may attain a condition of very variable temperature, causing 
irregular refraction of rays of light. Objects viewed through such 
an atmosphere appear distorted and of changing form. The 
graduations of the staff appear to be contracting, expanding, and 
wriggling in a confusing manner so that readings can only be 
approximately guessed at and accurate levelling is out of the 
question. If the work is important the levelling should be delayed 
till the atmosphere has attained a more uniform temperature. 

Variable heating of the tei-scope and head of the level may cause 
temporary warping and consequent faulty adjustment. The 
errors resulting from this may tend to be cumulative if the sides of 
the instrument are alternately exposed to the sun in taking back- 
sights and foresights. For important work the errors from this 
source may be avoided by keeping the level carefully shaded from 
the sun's rays. 

On frozen or ice-covered ground the iron-shod points of the legs 
are liable to a slight, gradual settlement under the weight of the 
instrument. The conditions are worst when a thaw has com- 
menced, and the amount of settlement of the level and of the staff 
may then be serious. Errors caused by settlement of the level 
and the staff are always in the same direction and are cumulative. 
When levelling requires to be carried on under the above-mentioned 
conditions the backsight and foresight for each position of the instru- 
ment should be taken as rapidly as possible after each other, so 
that the level gets little time to settle, and the change points 
should, where possible, be firm, permanent objects which are 
not ice-covered. 

Mistakes in Reducing the Levels. — In the work of computing the 
reduced levels, involving as it does a large amount of subtraction 
of figures in different columns, it is hardly surprising if arithmetical 
mistakes should occur pretty frequently. The permitting of these 
mistakes to pass undetected and to remain as errors in the work can, 
however, only be attributed to gross carelessness. Such errors may 
be almost entirely eliminated by the application of the appropriate 
checks, as described in Chapter XVI., pp. 247 and 249. 

Appropriate Length of Sight.— The distance at which a clear and 



258 SURVEYING 

precise reading of the staff can be made depends on the power of 
the telescope, the nature and distinctness of the graduations and 
numbering on the staff, and the condition of the light. In sunlight 
much longer sights can be taken when the view is towards the sunny 
side of the staff than when it is towards the shady side. It is seldom 
that precise readings can be made at distances much greater than 
600 ft. Long sights may be taken in rapid preliminary work, 
such as spot levelling, where speed is of importance and great 
accuracy is not required ; also to unimportant isolated points 
where the reading of a fairly long sight might save an additional 
setting of the level. 

Beginners are apt to have a leaning towards making the sights 
as long as possible, from an impression that the accuracy and 
rapidity of the work will be enhanced owing to the reduced number 
of changes required. From the foregoing discussion of the errors 
in levelling, however, it will have been seen that taking all sources 
together the error in a sight increases more rapidly with the length 
of the sight, so that the error in a single sight of 300 ft. will under 
similar conditions be less than half the error in a sight of 600 ft. 
A further reason for the employment of short sights lies in the fact 
that in practice the total error in levelling over a given distance 
does not mount up in direct proportion to the number of sights 
but more nearly in proportion to the square root of the number 
of sights. 

It is probable that for best accuracy the lengths of foresights and 
backsights should average about three chains or, say, 200 ft., and 
should not exceed 300 ft. or be much less than 100 ft. There is the 
further advantage in the adoption of short sights, that the observer 
at the level thereby has his assistants always within convenient 
distance for communication, observation, and control, and the 
work proceeds easily and smoothly and with the least chance of 
mistakes. 

Permissible Error in Levelling. — The degree of accuracy desirable 
in levelling work depends, among other things, on the nature of the 
work and the time and money available for its execution. At the 
one extreme we may place the rough preliminary work required 
for the location of a road or railway, where the endeavour is to 
obtain as rapidly as possible a broad determination of the features 



ERRORS IN LEVELLING 259 

of the country sufficient to locate a route. At the other extreme 
may be placed the very accurate work required in the setting out of 
works of the nature of drainage or water supply tunnels, where 
regular gradients may have to be laid out with a total fall of less 
than one foot per mile, or the precise work of the Ordnance Survey 
Department in levelling for the establishment of bench marks, &c, 
in this country. 

The error which accumulates in a stretch of levelling can only be 
exactly determined when a circuit is completed returning to the 
starting point. The total accumulated error is then the difference 
between the elevation assumed for the starting point at the com- 
mencement of the levelling and the elevation of the same point 
found by calculation as the result of the levelling. As already indi- 
cated, the error in levelling may be expected to accumulate, not in 
direct proportion to the number of changes, but more nearly as the 
square root of the number of changes. The number of changes 
per mile being fairly constant we may express the error by the 
formula 

E = CjB 

where E is the accumulated error in feet, D is the distance levelled 
in miles, and C is a constant for levelling of any particular degree 
of accuracy. The value of C may be taken as a measure of the 
precision of the levelling. 

With ordinary instruments and without special precautions an 
error of 1 in. or *08 ft. in a mile would represent fair accuracy. 
The error in four miles under the same conditions might be expected 
to be 2 ins. The formula expressing this degree of fair accuracy 
would be E= -08 Vd. 

Good ordinary work would be represented by the formula 
E = -05 Vd7 while E = -02 vD would represent very precise 
work, such as could only be attained by the use of a special instru- 
ment and the adoption of special precautions. 



82 



CHAPTER XVIII 

SECTIONS, CONTOURS, ETC. 

This chapter deals with levelling applied to particular purposes 
such as the taking of longitudinal and cross sections and the obtain- 
ing and locating of contour lines. Consideration is also given to 
methods and devices which may be used in particular cases, such ' 
as levelling up a steep slope, levelling over summits and hollows, 
taking levels of overhead points, levelling past obstructions of 
various kinds, levelling by the reciprocal method. 



Longitudinal and Cross Sections. — In taking a longitudinal 
section along a line marked out on the ground, the elevations 
of a series of points on the line are determined by levelling, the 
positions of the points being simultaneously located by chaining 
along the line and noting their distances from the point of com- 
mencement. Where the vertical profile of the ground is regular or 
gradually curving levels will be taken on points at equal distances 
apart and generally at intervals of a chain length. On irregular 
ground, that is, where abrupt changes of slope occur, the points 
should be chosen in the positions best suited to accurately deter- 
mine the section. This will be accomplished by taking the levels of 
all points where the slope of the ground changes, in addition to the 
levels at each complete chain length. 

In the designing of works which occupy only a narrow strip of 
ground, such as sewers, water conduits, &c, a longitudinal section 
along the centre line of the track gives all the information that is 
required as to the surface of the ground. In the case of works 
which occupy a strip of ground of some width, such as railways, 
roads, &c, a longitudinal section along the centre line will serve the 
purpose if the ground is level across in the direction at right angles 
to the centre line. But if the ground has a variable cross slope, 
the information given by the longitudinal section will not be 






SECTIONS, CONTOURS, ETC. 261 

sufficient and must be supplemented by means of cross-sections. 
These are short sections taken at intervals at right angles to the 
chain line and extending usually a little way beyond the limits of 
the intended works on either side. In the case of works which are 
to occupy a broad area of ground the required information as to the 
features of the surface will usually be obtained by a series of parallel 
cross-sections taken at right angles to a base line or by several 
groups of such cross-sections taken from different base lines, and 
arranged to cover the area in the manner best suited to the 
circumstances. 

Longitudinal Sections. 

To run a longitudinal section expeditiously a party of four 
persons is required, namely, a leader of the party who will work 
the instrument, book the readings, and direct the operations, a 
staffholder, and two chainmen. By making one of the chainmen 
act as staffholder the party may be reduced to three persons, 
while at a pinch two men can do the work, in which case the opera- 
tions of chaining and levelling are carried on alternately. 

As a rule the levelling will be arranged to start from a bench mark 
or from some point of known elevation, and will proceed by alternate 
backsights and foresights till the level is set up in a suitable posi- 
tion to command the commencement of the longitudinal section. 
Chaining will then proceed and the staffholder will hold on the 
points at distances, 0, 100, 200, 300, &c, and on such other inter- 
mediate points as are required to give an accurate section, or as 
directed by the chief of the party. The points will be given in 
the order in which they occur along the section, and the staffholder 
will call out their distances to the observer at the level, who will 
book them along with the staff readings. When the limiting length 
of sight is reached, or when it becomes necessary to shift the level 
owing to other circumstances, such as the line of sight passing 
entirely above or below the staff, or the view becoming obstructed, 
a change point will be taken which may either be a point on the 
section, if firm enough, or some other point specially chosen, and 
the foresight will be read. The precautions of swaying the staff, 
and checking the reading after it is booked, should be observed. The 
level may then be lifted, carried forward and planted in a suitable 
position from which to overtake the next portion of the section. 



262 SURVEYING 

The backsight will be read as carefully as the foresight and the 
chaining and levelling of the section will then proceed as before. 

For purposes of checking and future reference, temporary bench 
marks should be established at intervals along the section, say, 
three or four to each mile. Should any bench mark or point of 
known elevation occur near the section line the opportunity should 
be taken of checking on to it, and working out its reduced level 
from the booked figures there and then. If this does not agree with 
its previously known elevation, a line of check levels should be run 
backwards to the point of commencement, picking up on the way 
any temporary bench marks which may have been established. 
When the reduced levels of these bench marks and of the com- 
mencing point have been calculated from the check levels a com- 
parison with their first determined elevations will show whether any 
mistake has occurred, and where. If there is a mistake, it will be 
located as occurring somewhere between two of the bench marks, 
and the portion of the section affected can then be levelled over 
again. 

Where there are no existing bench marks and only one levelling 
party is working, the field work will be checked by the method of 
levelling back to the starting point. A long section would be taken 
in portions, each portion being checked before the next is levelled, 
so as to minimise the amount of wasted labour in case a mistake 
should occur. It is a good plan to arrange the operations so that 
each day's work stands completely checked by itself. 

If speed is important two levelling parties may be employed, 
one running the section and the other checking, or one party may 
go forward establishing bench marks, which will be used as checks by 
the second party running the section, or two parties engaged on 
different work in the same vicinity may agree to meet at certain 
pre-arranged places and check each other by levelling on to the 
same point. 

Example of a Section for a Small Sewer. — Plate III. shows a 
longitudinal section for a small branch sewer. The portion of the 
level book referring to this section is given on pp. 264 and 265. 
The levelling is commenced from an Ordnance bench mark, and 
finishes for purposes of checking on a temporary bench mark which 
had previously been established in connection with another portion 



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SECTIONS, CONTOURS, ETC. 263 

of the same work. In chaining the section distances have been noted 
to a few easily recognisable points, such as cuts of side streets, 
lines of prominent buildings, &c, so that the position of the sewer 
may be accurately laid down on the Ordnance Survey map, which 
was utilised in preparing the plan of the sewers. 

In plotting a longitudinal section the vertical heights will usually 
be drawn to a much larger scale than the horizontal distances, unless 
the natural slopes of the ground are steep. In the section illustrated 
the horizontal scale employed was 1 in. = 60 ft., and the vertical 
scale 1 in. = 10 ft., giving a ratio of exaggeration of six. The 
horizontal distances of all points at which levels have been taken 
are first marked off along a datum line chosen to represent some 
convenient elevation in round figures, and such as will give a con- 
venient height of section when plotted, having regard to the amount 
and nature of the information requiring to be marked thereon. 
Perpendicular lines are then drawn in pencil through these points 
and the vertical heights above the datum line are marked off to 
scale giving a series of points on the profile of the ground. A 
continuous line drawn through these points gives a representation 
of the surface of the ground to an exaggerated vertical scale. 

If a complete record of the section is desired all the vertical lines 
used in plotting may be inked in, and the corresponding elevations 
of the surface written in ink alongside. In the section illustrated 
only such information has been recorded as might be required in 
the setting out of the works, or in taking out quantities, or for future 
reference. 

In finishing a working section such as this it is customary to show 
in red the intended new works and all notes and information referring 
to them, the rest of the section being shown in black. 

Example of a Longitudinal Section for a Railway. — Plate IV. 
shows a portion of a working section for a double line railway, 
and Plate V. shows the corresponding portion of the working 
plan. 

In connection with the promotion of a parliamentary Bill for a 
railway in Britain it is a requirement of the standing orders 
that the distances on the plans and sections shall be given in miles, 
furlongs and chains (of 66 ft.). The same system of measurement 
is usually (but not necessarily) adhered to in preparing the working 



264 



SURVEYING 



plans and sections, and is the system adopted for the section 
illustrated. The vertical heights are in feet. 



f ,3 

Seu/er in (Vain Street 






Inst. 
Height 


Back 
Sight 


Inter- 
mediate 


Fore 
Sight 


Reduced 
Levels 


Distance 




4G3 


94 


/ 


84 










462 


10 














// 


40 






452 


54 


100 
























13 1 












// 


35 






452 


59 


150 












10 


30 






453 


55 


200 












8 


48 






455 


46 


250 












5 


87 






458 


07 


300 












4 


eo 






459 


04 


32+ 
























334 












2 


le 






461 


75 


400 












7 1 


50 






46Z 


44 


43/ 




469 


38 


5 


45 









Of 


463 


93 














5 


20 






464 


18 


500 












4 


88 






464 


50 


5IO 












3 


37 






466 


Of 


600 












3 


30 






466 


08 


636 












5 


02 






464 


36 


700 












5 


82 






463 


56 


757 












5 


73 






463 


65 


800 












5 


05 






464 


33 


858 
























890 












5 


35 






464 


03 


eoo 












5 


77 






463 


61 1 94 














4 


67 


464 


7/1 








































1 


v 





















For this section the centre line of the railway was set out with 
the theodolite, and pegs were driven at each chain from the com- 



SECTIONS, CONTOURS, ETC. 



265 



mencement. The levelling of the section followed the staking out, 
the staff being held on each peg and where necessary on intermediate 



( Sect/on Q°i n $ South ) 



Description 



O.B.m. On Church 4-62 / 



Line of centre of 



passaqt 



S. Line of High Street 



Centre of 3 road tie Road 



Tu 



rn . Line of projecting house £. '. side 



S . Gable . House nearest" roocd on W. side 



Line of front garden ujg/f £. side 



Opp. AtfJ2± 



S- Gab/e Co-op. 3ccitd/nqs 



Turn opp- S. pate - post iast Mouse £.s/de 
T. B '. M. On stone at corner of ra/'/inq 
fast' house <4e4'68 



points also. The levels of the latter are made use of in plotting 
the surface line but are not recorded on the section. 



I 



HUH""**** 1 **""* 1 *"*"*' 






.sHHiHtM*" 2 



Plate IV. Portion of Working Longitudinal Section of Kailu 

#K * 111* 

^* S S S 55 I 1 1 1 § § 1 S S* * a a a *T 














s s < 



i 1 1 



M 



: 






u 



865/ts.X t| k ySSSIhs. 



:*OChs. R.H. ■ 
20Chains. 



266 SURVEYING 

The horizontal scale of the section is the same as the scale of the 
plan, namely 1/2500 or 1 in. = 208-33 ft. ; the vertical scale is 
1 in. = 30 ft. The datum is Ordnance datum, that is, mean sea 
level at Liverpool. 

Of the two parallel lines drawn in the upper portion of the section 
the lower one represents the formation level of the railway, that is, 
the finished surface of the earthwork on which the ballast is to be 
placed. The upper line represents the level of the top surface of the 
rails. Formation level is generally shown by a red line and rail 
level by a blue line. 

The information to be recorded on the section comprises the 
following : — 

(a) The names of important objects or features crossed by the 
railway, such as roads, streams, railways, &c. These are printed 
above the section in line with the objects to which they refer. 

(b) The ground surface level at each peg. These levels are 
written in black vertically along the lines to which they refer, and 
form the bottom row of figures on the section. 

(c) The formation level at each peg. When the gradients have 
been fixed and the formation line has been drawn on the section 
the formation level at each peg is calculated, working from the levels 
which have been established at the changes of gradient. The 
formation levels are usually written in red ink immediately above 
the ground surface levels. A thick vertical red line marks the 
position of each change of gradient, and the formation levels at 
the changes are printed in bold figures so as to stand out con- 
spicuously from the others. 

(d ) The depth of cutting or height of embankment at each peg. 
These depths and heights are found by taking the difference between 
the formation level and the ground surface level at each peg. The 
result will be cutting or embankment according as the formation 
level is smaller or larger than the ground surface level. The 
calculated depths and heights should agree throughout with the 
corresponding dimensions scaled from the section. 

(e) The gradients of the railway. These are printed boldly in 
red above the row of formation levels. 

(/) The distances along the section. These are printed under- 
neath the datum line. In the section illustrated the distances are 
marked in figures at ten-chain intervals (660 ft.) giving miles and 



Plate IV. Portion of Working Longitudinal Section of Railway. 



Sfi. 

- -9 *■* 



^18 



fp cm 6 o5 as- 

tl "S ? E*^ * *> •* 

a c\i ~- -S ~. 



i o) oi 




I 



S^§! 



5 5 | ^ 5 § 



^?5S 






Vertical Scale 
100 

\ ' ' ' ~ r 



Rad. 60Chs.L.H. 
150 



M5-6lks. 



SECTIONS, CONTOURS, ETC. 267 

furlongs from the commencement of the line. Where the pegs 
have been placed at 100 ft. intervals they are usually numbered 
consecutively, and instead of distances, the number of each tenth 
peg from the commencement is printed on the section. Peg No. 237 
would thus be at a distance of 23,700 ft. from the starting point of 
the section. 

(g) A note of the datum used, a descriptive title of the section, 
and the vertical and horizontal scales to which the work is plotted, 
accurately drawn once or oftener, according to the length of the 
section. 

In addition to the above information a railway section should 
also show the position of all separate items of work, such as bridges, 
culverts, &c, with a note of their leading dimensions and a reference 
to the numbers of the drawings on which they are detailed. In the 
case of a bridge carrying the railway over a road the note on the 
section would be similar to the following : " Underb ridge, Span 
40 ft., Headway 16 ft., Drawing No. 37." 

The positions and results of any borings or trial pits which have 
been made to determine the nature of the materials in the cuttings 
will also usually be shown on the section. 

For general information and as a guide in the arranging of the 
gradients the positions and radii of the curves of the railway may be 
noted underneath the section as in the example illustrated. 

Cross-Sections. 

Cross-sections are made use of in connection with the construc- 
tion of railways, roads, canals and works of like nature for two 
principal purposes, namely : — 

(a) To determine the area of ground covered by the works. 

(b) To determine the quantities of excavation, embankment, &c, 
in the earthworks. 

For rough purposes short cross-sections may be taken with an 
inclinometer or Abney level. If the ground has a uniform slope at 
right angles to the longitudinal section, the information requiring 
to be noted in the level book is the angle of the slope and the 
direction of the fall, whether towards the right hand or towards 
the left hand. Right and left hand in connection with cross-sections 
have reference to an observer standing on the longitudinal section 
and looking forward in the direction in which it is being run. It 



268 



SURVEYING 



is important in sketching and plotting cross-sections to see that the 
slope is never laid off to the wrong hand. Inclinometer cross- 



CrOSJ* Sechnnj* Ayr Rcxi/u/ay Af?3 






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sections may be booked by sketching in a manner similar to that 
shown on p. 269. 




3&H 




SECTIONS, CONTOURS, ETC. 



269 



An adaptation of the mechanic's level furnishes a useful method 
for rapidly taking cross-sections on steep ground. A rod 5 ft. in 



>k 












Description 








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height is used with a cross-piece and pair of sights fixed on top 
accurately at right angles. A mechanic's level is attached to the 



270 SURVEYING 

cross-piece or laid on it, and indicates when the line of the sights 
is level. A mirror may be fixed so that the bubble may be seen while 
a sight is being taken. In taking a cross-section the rod is held on 
a levelled point of the longitudinal section with the cross-piece in 
the line of the cross-section. When the bubble is at the centre a 
sight is taken with the eye, and an assistant is directed to mark the 
point where the line of sight strikes the ground. This point is 
5 ft. above the point on the longitudinal section, and the horizontal 
distance is measured with the tape. The appliance is brought up 
and held on this point and the taking of the cross-section is continued 
to the desired distance by successive vertical steps of 5 ft. and 
measured horizontal distances. 

The above description applies to the taking of a cross-section 
working uphill. In proceeding downhill from the longitudinal 
section the appliance requires to be set up and a few trial sights 
taken till a point is found from which the line of sight strikes the 
higher point. 

Cross-sections taken by this method in vertical steps of 5 ft. are 
very easily and rapidly plotted on cross-section paper. 

If the cross-sections extend to some length and require to be 
accurate they must be taken with the level. In this method the 
levels of the points on the cross-sections are calculated in the usual 
manner and the cross-sections are plotted from a datum line as in 
the case of the longitudinal section, although they may also be 
plotted directly from the staff readings in the manner indicated 
below. 

A concise graphical method of booking cross-sections which 
greatly facilitates plotting is illustrated on p. 269. The levels of 
the pegs and intermediate points on the longitudinal section are 
booked in the ordinary way, but for the cross-sections the staff 
reading on each point and the distance of the point from the centre- 
line peg are marked on a sketch. In plotting the cross-sections a 
separate base line is used for each, the points being plotted by laying 
off the staff readings downwards from the base line, as shown in 
Fig. 215. ABC is the temporary base fine from which the ground 
surface line A'B'C is plotted by laying off the staff readings 2-60, 
6*79, &c., vertically downwards to scale at the points at which the 
levels were taken. Point B' represents the centre-line peg, and its 
level is written on the cross-section to serve as a datum for the 



SECTIONS, CONTOUKS, ETC. 271 

plotting of the formation level, when the latter has been worked out 
from the longitudinal section. The arrangement of a series of 
cross-sections when plotted in this manner is shown in Fig. 216. 
A line representing the centre line of the railway is drawn either 
vertically on a sheet of paper of the usual size or along the middle of 
a narrow roll. The cross-sections are plotted at close intervals 
about the centre line, sufficient space being always allowed for the 
plotting of the embankment or cutting on each. The cross-sections 
are best plotted in consecutive order from the bottom to the top of 
the sheet or from left to right in the case of a continuous roll. 

Long cross-sections will, as a rule, be levelled and booked in the 
manner described for longitudinal sections. 

The labour of plotting cross-sections is diminished and the speed 
increased by the use of squared paper, known as " cross-section 
paper." The two sets of lines crossing each other at right angles 
are usually spaced either 

ten or twenty to the lineal A - B C 

inch, every fifth or tenth 
line being drawn heavier 
than the others as a guide 
in counting the spaces. 
The scale chosen for the 

cross-sections is made to suit the divisions of the paper. Cross- 
section paper is specially useful in plotting work directly from the 
staff readings. 

Contoues. 

A contour is an imaginary line on the surface of the ground 
passing continuously through points of equal elevation. The 
water edge of a still lake is a contour fine, and a contour fine at 
any given elevation may be imagined as the shore line formed by 
still water flooding the earth up to that elevation. 

Use of Contours.— Contours are usually plotted on survey maps 
and engineering plans at equal vertical height intervals. The 
interval may be 1, 2, 5, or 10 ft., or some such amount, in the case 
of plans to be used for purposes of designing and laying out of works 
and calculating quantities, while in the case of topographic maps 
giving a view of an extensive territory the interval may be 10, 20, 
50, 100 or 200 ft., or more, depending on the scale and purpose of 



: 




— ■» — 


~~C 


A' 


Fig. 215.- 


Y / 






—Plotting Cross-Section. 





272 



SURVEYING 




Fig. 216.— Series of Cross-Sections. 



SECTIONS, CONTOURS, ETC. 



273 



the map. Contours plotted at equal height intervals show the form 
of the ground surface very graphically. The main points to be 
noted in reading the information conveyed by a contour plan are 
illustrated in Fig. 217. 




Fig. 217.— Typical Contour Plan. 

The contours being at equal height intervals the steepness of the 
slope of the ground will be proportional to the closeness of the spacing 
of the contours on the plan. Thus in the figure the steepest slope 




Fig. 218.— Section Plotted from Contour Plan. 

is shown at A while at B the ground is comparatively flat. Uniform 
spacing of the contours represents a slope of uniform gradient as 
at C, and if in addition the contours are straight the slope will be 
a plane surface. The direction of maximum steepness at any point 
is at'right angles to the contour as represented by the arrow at C. 
Hills are represented by a series of closed contours of diminishing 
s. T 



274 SURVEYING 

size and increasing altitude lying within each other as at D and E. 
A similar series with the altitude diminishing towards the smallest 
contour ring would represent a depression. 

Valleys are shown at F, G and H, where the contour lines converge 
on either side towards the stream or valley bottom. Each contour 
thus forms a more or less acute angle pointing towards the head 
of the valley, and the elevation of the contours increases as you 
proceed in the direction in which the angles point. 

A ridge, as shown by the dotted line KD, is practically the con- 
verse of a valley. The contours form a series of rounded angles 
pointing in a general direction along a straight or curving line, 
and taken in order in the direction in which they point they are of 
diminishing elevation. 

A watershed is shown by the dotted line DLE. It passes from hill 
to hill at right angles to the contour lines and through the highest 
point L of the pass between the hills. 

Fig. 218 shows a section along the line DE as plotted from the 
contours. To plot a section along any given line on a contour 
plan, a simple method of procedure is to lay the edge of a strip of 
paper along the line on the plan and mark off in pencil the points 
where it cuts the contours, and then transfer these points to the 
datum line of the section. Verticals are erected at each of 
these points, and the contour levels are then plotted giving points 
on the surface of the ground. Instead of plotting the level of each 
point separately, horizontal lines may be drawn at the levels of the 
contours, as shown in Fig. 218, the points on the surface being then 
given by the intersection of each vertical with its proper contour 
level. 

The heights of the contours should be marked on the plans, as 
indicated in Fig. 217. It is not necessary to figure the level of every 
contour, but usually every second or fifth will be marked, the figures 
being arranged in continuous lines suitably distributed over the plan. 
The clearness of the plan will be enhanced by emphasising every 
fifth or tenth contour, either by drawing them in a heavy line, 
as in Fig. 217, or by using ink of a different colour. The contours 
emphasised will be those whose levels are the roundest multiples 
of 10. 

The contours require to be carefully figured where depressions 
occur. Fig. 219 shows the contours of a conical hill with a volcanic 



SECTIONS, CONTOURS, ETC. 



275 



crater on its summit, the true section across the summit being shown 
in Fig. 220. The hill might be taken to be of the form shown in 
Fig. 221, if the level of each contour near the summit were not 
carefully marked. 

1700 



Pig. 219. 




Fig. 220. 



Fig. 221. 



Figs. 219, 220, 221.— Contour Plan and Sections 
of Volcanic Crater. 



Some of the purposes for which contour plans are useful are the 
following : — • 

(a) Location of final routes of railways, roads, &c, and in the 
location and arrangement of works of engineering generally. 

(b) Town planning and laying out of building areas. 

(c) Calculation of capacity of reservoir. 

(d) Calculation of quantities of earthwork. 

Contour maps are useful as showing comprehensively the features 



276 SURVEYING 

of elevation of a large area, and for purposes of prospecting, locating 
approximate routes, determining extent of drainage areas, &c. 

Locating Contours. — (a) By marking the contours out on the 
ground and then surveying them. 

One method of procedure is to first set out a series of pegs at the 
contour heights along a line running, if possible, from the lowest 
ground to the highest, and then use these pegs as the starting points 
from which to set out points on the actual contours on the ground. 
On a large area several such series of pegs would be set out, and in 
any case it is advisable to have at least two lines, so that the 
accuracy of the contours may be checked by having them start on a 
peg on one line and finish on a peg at the same level on the other line. 

In setting out the lines of pegs and the contours, the levels must 
be reduced as the work proceeds, as the height of instrument at 
each set-up requires to be known at the time. Assume that 
the levelling has commenced from a bench mark and that the 
instrument is now set up in position to commence the setting out of a 
line of pegs. The levels have been reduced, and the instrument height 
is found to be 58-34. Pegs are to be set out at vertical intervals of 
5 ft., starting from elevation 50. To set the first peg at elevation 50 
the staff must read 8-34 and the staff-holder is, therefore, directed 
up or down hill till the staff held on the ground gives this reading 
very nearly. The peg is then driven partly down and, finally, a 
little at a time, till the staff held on its top gives the correct reading. 
The peg at contour 55 may be got at the same setting of the level 
by driving it to give the reading 3-34 on the staff. The instrument 
would then be shifted uphill to a suitable position for setting out the 
peg at elevation 60 and one or two higher ones if they were within 
horizontal range. The instrument height is worked out for this 
position, and by subtracting 60 from it the staff reading for setting 
the peg at contour 60 is obtained, and so on for the other pegs. 

To run a contour on the ground the level is set up in a suitable 
position to read on to the peg at the level of that contour, and the 
reading is taken on the staff held on the peg. This gives the height 
of the instrument above the contour and, therefore, any point on the 
ground which gives the same staff reading will lie on the contour. 
The staff -holder, therefore, walks along the contour as nearly as he 
can judge for, say, twenty or thirty paces and holds the staff on the 



SECTIONS, CONTOURS, ETC. 277 

ground. The observer at the level notes the reading, and directs 
the staff-holder uphill or downhill accordingly until the proper 
reading is obtained. The exact spot is then best marked for future 
location by sticking a piece of lath into the ground. A series of 
points is set out in this manner within the range of the instrument 
and a change point is then taken, the instrument is moved forward, 
the new instrument height is calculated from which the staff reading 
for the second series of points is obtained, and these are then set 
out and marked by laths. The running of the contour continues in 
this manner to any desired length, or until it checks on to another 
peg or completes a circuit and returns to the starting peg. 

Where the contours are at close vertical intervals, and likewise 
at fairly close horizontal intervals, portions of two or more may be 
marked out at each setting of the level. 

Where there is any likelihood of confusion in surveying the 
separate contours, owing to the closeness of the spacing, it is advis- 
able to have several sets of laths painted in different colours or 
otherwise distinctively marked, and to run each single contour of 
one colour or marking throughout. 

The same principles should govern the spacing of contour points as 
apply in the case of surveying a boundary. Where a portion of the 
contour is evidently a straight line on the ground it will suffice to 
mark the end points of that portion. Where the contour is curving 
somewhat regularly, points will be taken at about equal intervals 
measured by pacing. The points must be taken close together 
where the contour is bending sharply, and all abrupt changes of 
direction should be carefully located. For a large scale plan the 
points to locate the contours should throughout be taken at closer 
intervals than for a small scale plan. 

The surveying of the contours after they have been run out might 
be effected by any of the methods of surveying which have already 
been described, and which may be suitable to the circumstances. 
The surveying of contours on the ground would, however, usually 
only be undertaken where they occurred at wide intervals, 
and method (6) next described would generally be preferable 
for the location of contours. 

(b) By taking spot levels or cross-sections, locating their positions, 
and subsequently, in the office, deducing and plotting the positions 
of the contours. 



278 



SURVEYING 



Fig. 222 shows an area with the cross-sections and levels which have 
been taken for the purpose of plotting the contours. The levels of 

the points may be 
o^ ^ ^ marked simply in 

pencil. The con- 
tours will seldom 
happen to pass 
through any of 
the points whose 
levels have been 
taken. The posi- 
tion of a contour 
between two 
levelled points is 
fixed by assum- 
ing that the 
ground has a uni- 
form gradient be- 
tween the points. 
The contour must 
then divide the 
horizontal d i s - 
tance between the 
points in the same 
proportion as it 
divides the 
vertical distance. 
Where the dis- 
tances between 
points on the plan 
are not great the 
positions of the 
contours are usu- 
ally interpolated 
by estimation. 
An accurate method of finding the positions of the contours on 
the cross-sections is by plotting the latter and drawing horizontal 
lines at the elevations of the contours. The intersections of these 
horizontal lines with the ground surface line give the positions of 




SECTIONS, CONTOURS, ETC. 



279 




Fig. 223. 



d' 
C.S.N 5 
-Finding Positions of Contours. 



the contours, and these may be transferred to the plan. This method 
is illustrated in Fig. 223, which shows cross-section No. 5 on the 
preceding figure plotted from the levels. Points a, b, c, d, &c, 
where the contour 
elevations meet the 
ground line, enable 
the corresponding posi- 
tions of the contours 
to be plotted on the 
plan. 

Contours may be 
rapidly and accurately 
interpolated between 
the levelled points by 
the employment of 
graphical methods. In the method illustrated in Fig. 224, a system 
of equally spaced parallel lines is drawn on tracing cloth, every tenth 
line being emphasised. The thick lines on this diagram are num- 
bered to represent contours with an interval of 1 ft. It is required 
to interpolate the contours between two points, A and B, whose 
elevations are respectively 111-4 and 114-3. The thin line numbered 
1-4 on the diagram would be placed over point A. the tracing would 

be held down with 
the finger or with a 
pricker at point A 
and rotated till the 
fine numbered 4-3 
passes through the 
point B. It will be 
evident that the 
thick lines numbered 
2, 3, and 4, then cut 
the line joining A 
and B in the posi- 
tions of the contours 
112, 113, and 114, and these points may be pricked through on to 
the plan. 

The lines on a diagram, such as the above, need only be tem- 
porarily numbered in pencil to suit the particular work in hand. 




Fig. 224. — Finding Positions of Contours. 



280 



SURVEYING 



For large scale plans the heavy lines may represent single feet, 
while for smaller scales they may represent intervals of 2, 5, 10 or 
more feet, as may be found convenient in plotting. 

Another graphical device employing lines drawn on tracing cloth 
is illustrated in Fig. 225. There is a radiating system of lines with 
every tenth line emphasised, and crossing this system there is a 
series of parallel guide lines. In using the diagram the guide lines 
must be kept parallel to the line joining the two points on the plan. 
To place it in position for giving the contours between the points 



% Hlfg §^= ^^ ^^ ^^: ^^ 
J §§111 ^§ |gll|ilEifcil= ===== 



Fig. 225. — Finding Positions of Contours. 

C and D, the line numbered 2-5 would be placed over point C and the 
diagram disposed so that the guide lines are parallel to the line CD. 
The diagram would then be moved to the right or left without 
rotation, keeping line No. 2-5 over point C, till the line No. 4-9 
passes through point D. Line CD will then be intersected by the 
heavy lines of the diagram in the positions of the contours. 

Use of Contour Plans. — (a) Location of route. A simple case of 
the use of a contour plan in the designing and location of engineering 
works is illustrated in Fig. 226. The problem is to lay down 
on the plan from point A a suitable route for an outfall sewer 



SECTIONS, CONTOURS, ETC. 



281 




282 SURVEYING 

with a gradient of 1 in 300 and a minimum depth of 6 ft. to 
the invert of the pipe. The sewer is to be in straight lengths 
between manways which are to be formed at intervals of about 
100 yards. 

The level of the invert at A is 133 ft. above datum and the level 
of the ground surface is about 141, so that the depth at this point 
is about 8 ft., that is, 2 ft. more than the desired depth of the outfall 
sewer. The first thing to be done is, therefore, to arrange the posi- 
tion of the first manway, at a distance of about 100 yards from the 
junction chamber, so that it will have a depth of about 6 ft. In a 
distance of 300 ft. the sewer will have a fall of 1 ft., so that the invert 
level at the first manway will be 132, and the ground surface level 
should, therefore, be about 132 + 6 = 138. If we, therefore, take 
a distance of 300 ft. on the dividers, and, placing one end at A find 
where the other point cuts the 138 contour line (point 6) this will 
fix the probable position of the manway. 

The contours are at vertical intervals of 2 ft. A fall of 2 ft. in 
the sewer will take place in a distance of 600 ft., so that if an ideal 
route were set out with a depth everywhere of exactly 6 ft. the line 
of the sewer would cut the contour lines at uniform intervals of 
600 ft. An approximation to the ideal route will be obtained if we 
take a distance of 600 ft. on the dividers to the scale of the plan 
and, starting from point b, step successively from contour to contour, 
thus fixing points c, d, e, f. The ideal route may now be sketched 
in, and is represented by the dotted line on the plan. The actual 
route to satisfy the requirements of the case will be a succession of 
straight lines approximating as closely as may be to the ideal route 
and arranged with manways at all changes of direction and at other 
intermediate points where necessary. The full line on the plan 
shows the line of the sewer as designed, the small squares indicating 
the positions of the manways. 

(b) Laying out of building areas, &c. The use of a contour plan 
is almost a necessity if the roads and streets of a proposed building 
area are to be laid out to the best advantage. The question of 
suitable drainage arrangements for the area is intimately connected 
with the scheme of lay-out of the roads, so that the features of eleva- 
tion of the entire area must be considered. A contour plan gives 
the information required as to these features, and when the contours 
are at equal vertical intervals and fairly close together we have the 



SECTIONS, CONTOURS, ETC. 283 

information in a very expressive form. The level portions of the 
ground, steep slopes, summits of hills, bottoms of valleys, and 
direction of natural drainage are all seen at a glance. The good or 
bad features of any proposed arrangement of roads will also be 
evident if a tracing of the scheme, prepared to the scale of the 
contour plan, is applied over the latter. When an arrangement 
of roads has been adopted, longitudinal sections of the roads may 
be at once plotted from the contours with sufficient accuracy to 
enable the gradients to be finally fixed, the amounts of cutting and 
embankment to be approximately determined, and the sewers and 
drainage arrangements to be designed. 

(c) Calculation of quantities. To illustrate the use of contours 
in the calculation of quantities we shall take the case of a reservoir 
formed behind an earthen embankment which closes in a valley. 
Contours of the area which will be flooded have been plotted on a 
plan at vertical intervals of 5 ft. Every contour which occurs 
below the level of the top of the embankment in this area will form 
a circuit passing along the inner face of the embankment and up 
both sides of the valley to meet on the line of the stream. We may 
imagine a series of level surfaces at vertical intervals of 5 ft. passing 
through the contours and dividing up the capacity of the reservoir 
into horizontal slices 5 ft. thick. The areas of the upper and lower 
surfaces of any slice will be the areas contained within the bounding 
contours, and the volume of the slice will be approximately obtained 
by multiplying the mean of the two areas by the thickness of the 
slice. The total capacity of the reservoir will be got by adding 
together the capacities of the separate slices obtained in this way. 
The capacity of the reservoir when the water is standing at any 
given level is required as well as the total capacity, and in the 
example on p. 284, the figures are arranged so as to give this 
information. The bottom of the reservoir is below the contour 
level 145, but the calculation of the capacity commences at level 
150, being the lowest point from which water can be drawn off 
for use. 

The contour areas may be calculated by the method of sub-division 
into geometrical figures, but in a case like this where we have a 
series of concentric areas with irregular outlines the employment 
of the planimeter will furnish the most expeditious method, pro- 
vided the plan is not large. 



284 SURVEYING 

Calculation of Capacity of Reservoir. 



Con- 




Mean 


Capacity of 


Capacity of Reservoir from Elevation 


tour. 




Area. 


Layer. 


150 up to Elevation. 




Sq. Feet. 


Sq. Feet. 


Cub. Ft. 


Cub. Ft. 


Gallons. 




145 


11,240 












150 


75,380 


114,790 


573,950 








155 


154,200 


196,950 


984,750 


573,950 


3,587,000 


155 


160 


239,700 


281,050 


1,405,250 


1,558,700 


9,742,000 


160 


165 


322,400 


357,600 


1,788,000 


2,963,950 


18,524,000 


165 


170 


392,800 


425,100 


2,125,500 


4,751,950 


29,699,000 


170 


175 


457,400 


498,250 


2,491,250 


6,877,450 


42,984,000 


175 


180 


539,100 


587,400 


2,937,000 


9,368,700 


58,555,000 


180 


185 


635,700 


685,950 


3,429,750 


12,305,700 


76,910,000 


185 


190 


736,200 


788,700 


3,943,500 


15,735,450 


98,346,000 


190 


195 


841,200 






19,678,950 


122,993,000 


195 










19,678,950 









Levelling Problems and Devices. 
Levelling on Steep Slope. — In levelling up a steep slope, as illus- 
trated in Fig. 227, the backsights will usually read nearly to the 
top of the staff and the foresights nearly to the bottom, and if the 
level is set up approximately in line between the change points the 
backsights will be on the average about twice as long as the fore- 
sights, and, therefore, error due to curvature will, as already 
explained, accumulate. The difference in length between the back- 
sights and foresights and the liability to error may be greatly 
reduced by setting up the level, not in line between the change points 
but at a considerable distance to the side, at points such as G and F 
in plan (Fig. 228), instead of points E' and D'. 

Levelling over Summits and Hollows. — Time and labour may be 



SECTIONS, CONTOURS, ETC. 



285 



economised by taking the precaution of only setting up sufficiently 
high to see over a summit, as illustrated in Fig. 229, the natural 
inclination being to set up right on the summit, a proceeding which 




Figs. 227 and 228.— Levelling up Steep Slope, 
might necessitate an extra change. The corresponding precaution in 
crossing a hollow of setting up only sufficiently low to enable the 
levels of all the required points to be obtained, as shown in Fig. 230, 
should also be observed. 

Taking Level of an Overhead Point. — The level of an overhead 
point may be obtained. by holding the inverted staff on it, as shown 
in Fig. 231, and taking the 
reading, which will give the 
height of the point above the 
line of sight. The reading 
must, therefore, be added to 
the height of the instrument 
to give the reduced level of 
the point, and should be entered in the level book marked with a 
plus sign. So that there may be no opportunity for mistake it is 
well also to make a note on the description page that the staff has 
been held inverted. Thus, for the case shown, the entry in the 




Fig. 229. — Levelling over Summit. 



286 



SUKVEYING 




column of intermediate sights would be -f- 10*27, and the relative 
description on the right-hand page " on underside of north girder 
at centre, staff inverted." 

Obstructions. — Most obstacles of the nature of obstructions to 
the line of sight can be overcome by levelling round about, but a 

little ingenuity expended in 
devising a simple expedient 
will often economise time and 
labour. 

To level past an obstruction 
such as a close-boarded fence, 
which can neither be seen 
over nor seen through, we 
may either drive a nail through a board near the bottom so as to 
project on both sides, or we may pass an arrow or a knife blade 
through one of the joints. In each case we obtain two points at 
the same level, one on each side of the fence, and they may be 
considered as one change point. The staff will be held for the fore- 
sight on the nail, arrow, or blade on one side of the fence and the 
level and staff will then be taken round, the level will be set up, 
and the backsight will be taken to the staff held on the portion of 



Bottom of staff 



Fig. 230. — Levelling across Hollow. 




W/MWM 
Fig. 231.— Level of Overhead Point. 

the nail, &c, which projects inside. The readings will be booked 
as for a change point in the ordinary way. 

In the case of a high brick wall we may fix on a horizontal joint 
near the bottom as a bench mark. Its position can be identified 
by counting the number of courses down from the top, and the same 
joint on the other side of the wall can be found by counting down 
the same number of courses. These two points at the same level 



SECTIONS, CONTOUKS, ETC. 



287 



will then be considered as one change point, the foresight being 
taken to the staff held on the joint on one side of the wall and the 
backsight to the staff held on the same joint on the other side. 
In some cases it may be better to fix points at the same level by 




Fig. 232. — Levelling past 
High Wall. 



measuring down equal distances from the top of the wall or from 
the under edge of the cope on each side. 

The foregoing methods are not applicable to the case shown in 
Fig. 232. A method of booking the readings and measurements 
arid obtaining the levels in the case of such an obstruction occurring 
in a longitudinal section is given below. 



Back. 


Inter. 


Fore. 


Ht. of 
Inst. 


Reduced 
Level. 


Dist. 


Description. 


— 


4-95 


4-68 


243-66 


238-71 
238-98 

252-42 


1,300 
1,366 


At base of wall, on 

stone. 
13-44 above stone to 

top of wall T.B.M. 


4-72 


9-22 
8-89 


— 


257-14 


252-42 

247-92 
248-25 


1,368 
1,400 


On T.B.M. on top of 

wall. 
At base of parapet. 



Crossing Pond or Lake. — In crossing a pond or lake advantage 
mav be taken of the fact that the surface of still water is a level 



SURVEYING 



surface. A reading may, therefore, be taken to the staff held at 
the water level on one shore as a foresight and on transferring the 
level to the opposite side, a backsight may be taken to the water 
level at any point of the shore. For the purpose of reducing the 
levels the foresight and backsight may be considered as taken on the 
same change point. 

In the case of a flowing river levels may be continued from one 
side to the other in the above manner with little error, provided 
care is taken to choose a comparatively still stretch and to see that 
the water levels are taken at points directly opposite each other. 



Reciprocal Levelling. — This is a method of levelling by which the 
true difference of elevation of two points a considerable distance 




Fig. 233. — Keciprocal Levelling. 

apart may be obtained by two sets of observations. The process 
employed automatically corrects for error in adjustment of the level 
and for the effect of curvature and refraction. It is applicable to 
cases such as the crossing of a wide river, where there is no bridge 
to enable the levelling to be continued across in the ordinary 
manner. 

Referring to Fig. 233, readings are taken with the instrument 
set up at point 1 to the staffs held on points A and B. The level 
is then transferred to point 2, the distance 2B being arranged 
practically equal to the distance 1A, and readings are again taken 
to the staffs held at A and B. The readings taken from position 
1 will give a certain difference of elevation, say, D 1} between the 
two points, and the readings from position 2 will give another 
difference of elevation, say, D 2 . The true difference of elevation 



is the mean of the two determinations, that is 



2 



SECTIONS, CONTOURS, ETC. 289 

Suppose that the readings on the staffs were as given below — 

From Position 1. From Position 2. 

Reading on A .... 5-75 7-90 

B .... 4-32 6-33 

Diff. J) 1 = 1-43 Diff. D = 1-57 
I.43 _i_ 1 .57 
Actual difference of elevation = — = 1-50, that is, point 

B is 1-50 ft. higher than point A. 

Contour Grading. — The operation of running a line on the surface 
of the ground on a constant gradient along the face of sloping 
ground is known as " contour grading." It is frequently required 
in connection with the location of roads, railways, and water 
conduits. The operation may be performed with level and staff, 
in which case points would be marked on the ground at equal 
intervals in somewhat similar manner to that described for running 
a contour, the difference being that the successive staff readings 
from one setting of the instrument would not be equal, but would 
increase or decrease according to the gradient, thus on a falling 
contour gradient of 1 in 100, points might be marked at 100-ft. 
intervals, the staff reading being increased by 1 ft. at each interval. 
As in running contours, the levels require to be reduced as the work 



Contour grading will usually be more easily and expeditiously 
accomplished with the theodolite than with the level. The 
theodolite is set up on the ground on the line of the gradient at its 
commencement, and levelled and adjusted as for reading a vertical 
angle, the bubble of the telescope level being brought to the centre 
with the vertical circle index reading zero (see p. 35). The 
telescope is then tilted up or down to read a vertical angle equiva- 
lent to the gradient, thereby setting the line of sight to the required 
inclination. For gradients not steeper than 1 in 10 the vertical 

angle for a gradient — will be given very closely by the formula — 

n 

Vertical angle = minutes. 

n 

34-3R 
Thus for a gradient of 1 in 75 the angle would be = = 45'8 



290 SURVEYING 

minutes. The height of the telescope above the ground is taken 
on a staff or ranging rod, the telescope is sighted along the face of 
the slope in the direction in which the gradient is to run, and points 
are marked out as for an ordinary contour to give this constant 
reading on the staff. The reading is usually taken to the hand 
of the assistant held at the proper point on the staff or rod. 

The setting out of a contour gradient line from one position of 
the theodolite is continued only for such distance as the general 
direction of the line remains constant. Where the line takes a 
bend, and at other places where the conditions of view require it, 
a point will be set out on the gradient line, similar to a change 
point in levelling, the theodolite will be brought forward and set up 
over this point, and the setting out of the gradient line ahead will 
continue as before. 

The Abney level, described on p. 23, with the index set to the 
required angle, may be used in similar manner to the theodolite 
for rapidly running a rough trial gradient line. Headings are taken 
to a height of staff equal to the height of the eye above the ground. 
A contour gradient set out on the ground and marked with laths 
will usually appear as an irregular winding line. In the case of 
roads and water conduits, where curves of large radius are not 
required, the final centre line may often be set out directly from 
the contour gradient. Mean straight lines would be set out to 
replace those portions of the gradient line which keep a fairly 
constant general direction, and the various straights would be 
connected by curves of suitable radius. 

In the case of a railway, where the curves must always be of 
large radius (seldom less than 500 ft. for a very subsidiary line), 
the fixing of the best location for the centre line is not a simple 
matter. On hilly ground the usual procedure is to run a traverse 
survey, following the gradient line as closely as possible, and read 
the intersection angles. A section is taken with the level along the 
lines, and sufficient cross-sections are taken with the inclinometer 
or Abney level to determine the general form of the ground for some 
distance on either side of the traverse lines. The fixing of the best 
location for the centre line is subsequently a matter of office work. 
The traverse is plotted on paper, and from the cross-sections the 
actual contours are laid down. These enable a longitudinal section 
along any line to be plotted, and it may be necessary to plot sections 



SECTIONS, CONTOURS, ETC. 291 

on one or two trial lines before the most satisfactory location is 
attained. When a final centre line has been fixed on the paper, 
it is set'out on the ground, as regards the straights, by measure- 
ments from the traverse lines, the curves are run in by theodolite, 
as described in Chapter XIX., and accurate levelled longitudinal 
and cross sections are taken. These latter are used for the prepara- 
tion of the working plans. 



uz 



CHAPTER XIX 

SETTING OUT CURVES, ETC. 

In this chapter some methods of setting out circular curves on 
the ground will be considered. Methods involving the use of the 
chain and tape alone, or of the chain, tape and optical square, may- 
be adopted in certain simple cases where great accuracy is not neces- 
sary or where a theodolite is not at hand. For curves which extend 
to considerable lengths, and which require to be accurately set out, 
the use of the theodolite is almost imperative. The setting-out of 
building work is treated briefly. 



Methods requiring Use of Chain, Tape and Optical Square. — The 

following methods of setting out curves do not require the use of 
the theodolite : — 

(a) By offsets scaled from a plan. 

(b) By a radius swung from the centre of the curve. 

(c) By perpendicular offsets from a tangent line. 

(d) By offsets from chords produced. 

Method by Offsets scaled from a Plan. — This method is applicable 
to a circular curve or to any form of curve which has been accurately 
drawn on a plan to a sufficiently large scale. A base line or series 
of base lines is drawn on the plan to comply with the following con- 
ditions : The offsets should be as short as possible and the base 
lines should be chosen so as to be easily set out on the ground. The 
positions of the base fines are fixed by measurements scaled off the 
plan to definite points of existing objects which can be readily 
located on the ground, and by setting out these measurements on 
the ground a reproduction of the base lines as drawn on the plan 
will be obtained. Points are marked off by scale at equal distances 
along the base lines on the plan and the lengths of perpendicular 
offsets to the curve at these points are scaled off. These distances 



SETTING OUT CURVES, ETC. 



293 



and offsets being set out along and from the base lines on the ground, 
a series of points will be obtained lying on the required curve. 
This method would be applicable 
to the setting out of a scheme of 
winding roads for a park or pleasure 
ground where great accuracy as to 
position was not essential, or to any 
similar case. 




Fig. 234. 



—Setting out Curve 
by Eadius. 



Method by a Radius swung from 
the Centre of the Curve. — The general 
problem will be to connect up two 
straight lines tangentially by a circu- 
lar curve of given radius. In Fig. 
234, AB and DC represent the two 
straight lines which are to be joined up by a curve of radius R. The 
solution of the problem consists in finding point 0, the centre of the 
curve, on the ground. Set out the line EO parallel to line AB and 
at a distance R from it, and similarly set out FO parallel to and at 
distance R from DC. Point where the lines EO and FO 
intersect is the centre of the curve. The lines OB and OC set out 

at right angles to the lines AB 
and DC fix the tangent points 
B and C of the curve. The 
curve may be marked out on the 
ground between B and C by 
swinging a length of chain equal 
to radius R about the centre 0. 
The method is only useful for 
curves of very small radius. 

Method by Perpendicular Off- 
sets from a Tangent Line. — In 
Fig. 235, AB is a tangent to the 
circle AEF at the point A. C is 
the centre of the curve and CF 
is a radius parallel to AB. BE is the offset of length o from the 
tangent line to the curve at a point distant I from A, and DE is the 
corresponding offset of length p from the radius. 




Fig. 235.— Offsets to Curve from 
Tangent. 



294 



SURVEYING 



In the right-angled triangle CED we have 

p 2 = r 2_/2 

or p = Vr 2 — I 2 
But o + p = r or o = r — p. 
Therefore o = r — Vr 2 — I 2 

For ease of calculation the distances I should be taken in round 
numbers of units. A table of square roots will greatly expedite 
the work of calculation. 

Example. A portion of a curve of thirty chains radius is to be 
set out by perpendicular offsets at intervals of one chain from the 
tangent point. The figures for the calculation of the first eight 
offsets are shown in the following table : — 



r2 = 900. 




r = 30-0000 chs. 






J2 


r 2_?2 


Vr2 - 11 








r - A^ri-P 


Length of Offset. 


Chains 






Chains. 


Chains. 


Links. 


1 


1 


899 


29-9833 


0-0167 


1-67 


2 


4 


896 


29-9333 


0-0667 


6-67 


3 


9 


891 


29-8496 


0-1504 


15-04 


4 


16 


884 


29-7321 


0-2679 


26-79 


5 


25 


875 


29-5804 


0-4196 


41-96 


6 


36 


864 


29-3939 


0-6061 


60-61 


7 


49 


851 


29-1719 


0-8281 


82-81 


8 


64 


836 


28-9137 


1-0863 


108-63 



If the whole length of the curve to be set out forms only a very 
flat segment of a circle so that the last offset BC (Fig. 236) is not 
more than one-seventh of the length of I then the offsets may be set 
out as the ordinates of a parabola, and the resulting curve will 
hardly differ appreciably from a circle. The parabolic offsets are 
sometimes adopted in such a case because they are much more easily 
calculated than the circular offsets. For a parabolic curve the 
length of the offset is proportional to the square of the distance along 
the tangent from the tangent point or, 

Length of offset = cl 2 , c being a constant. 

Example : A straight line AB (Fig. 236) has been set out, and it 
is required to continue from the tangent point A along a curve 



SETTING OUT CURVES, ETC 



295 



which shall pass through point C. Lengths AB and BC measure 
463 ft. and 62*5 ft. respectively. The curve is to be set out by 
offsets at intervals of 50 ft. measured along the tangent from A. 

50 2 

Length of first offset o 1 = 62-5 X jw^ = 0-7289 ft. 

4:00 

The length of the first offset having been found as above the 
rest are worked out as in the following table : — 



o[ at 50 ft. = 0-7289 ft. 

2 „ 100 ft. = 0-7289 X 4 

3 „ 150 ft. = 0-7289 X 9 

4 „ 200 ft. = 0-7289 X 16 = o 2 x 4 


= 2-92 ft. 
= 6-56 „ 
- 11-66 „ 


o 5 „ 250 ft. = 0-7289 X 25 = o, x ^ 


= 18-22 „ 


o s „ 300 ft. = 0-7289 X 36 = o 8 X 4 

7 „ 350 ft. = 0-7289 X 49 

8 „ 400 ft. = 0-7289 X 64 = o 4 X 4 

9 „ 450 ft. = 0-7289 X 81 = o 3 X 9 


= 26-24 „ 
= 35-72 „ 
= 46-65 „ 
= 59-04 „ 


o l0 „ 463 ft. = 0-7289 X ^ 


= 62-5 „ 



It will be seen that most of the values can be obtained by short 
multiplication. 

The setting out of offsets becomes laborious and unsatisfactory 
when they exceed the length 
of the chain or tape. When 
the offsets for a circular curve 
become too long a new tan- 
gent line can be set out, as 
shown in Fig. 236. The line 
EC will be a tangent to the 
circle at C if the distance EC 
is equal to the distance AE. 
The point E will He a little 
beyond the centre of the 
length AB measuring from A 
and if D is the centre point 
the distance DE is equal to 




Fig. 236. 



D £ B 

—New Tangent. 



The point E being set out to 

this distance the line EC produced will give a new tangent line CF 
from which the same series of offsets may be set out as from the 



296 



SURVEYING 



tangent AB. Any required length ot curve can be set out from 
successive tangent lines in this manner. 

In the above offset method where equal distances are measured 

along the tangent line the 
pegs set out along the 
curve will be at varying 
distances apart. If it is 
required to set out points at 
equal intervals apart along 
the curve the offsets will 
require to occur at decreas- 
ing intervals from the tan- 
gent point. Referring to 
Fig. 237, A is the tangent 
point and is the centre of 
a curve of radius R. Points 
1, 2, &c., are to be set out at 
equal distances I along the 
curve by offsets from the 
tangent line. The calculation requires the use of trigonometry, and 
the first thing to do is to find the size of the angle a which an arc 
of length I subtends at the centre of the circle, that is, the angle 
AOl in the figure. 



- 













K 


""-•- 


"•-£ 




c 


\ 


\"7» 


x e 


f-'-'J? 


K • 














\' 






f i[ 


X, 


'•*''/ 


jo* 




t~" 


... 1 






rfi,—— 


— X, - 


>j 






•i — 




x 2 


->! 



Fig. 237. 



-Offsets at Equal Curve 
Intervals. 



Angle a = ^ radians = 



360 



R X 2tt 



_ I X 360 X 60 
~ R X 2 X 3 . 1416 

In the triangle BOl we have 



minutes 



X 3438 minutes. 



R 



= sin a, or x x = R sin a, 



similarly x 2 =. R sin 2a. 
Also we have p ± = R cos a, but o x -\- p t = R. 

Therefore o 1 = R — jh = R — R cos a = R (1 — cos a). 
similarly o 2 = R (1 — cos 2a). 

Example. Calculate the co-ordinates to set out six points at 
intervals of 100 ft. from the tangent point along a curve of 1,800 ft. 
radius. 



SETTING OUT CURVES, ETC. 
100 



297 



Central angle for arc of 100 ft. = r-g— X 3,438 minutes = 191 
minutes or 3° 11'. 



Calculation of Co-ordinates for Points equidistant along 
the Curve. 



LogE 


= 3-25527. 




Angle. 


Log Sin. 


Logs. 


X. 


ttl = 3° 11' 


8-74454 


1-99981 


100-0 = x x 


a 2 = 6° 22' 


9-04489 


2-30016 


199-6 = x 2 


a 3 = 9° 33' 


9-21987 


2-47514 


298-6 = x 8 


a 4 = 12° 44' 


9-34324 


2-59851 


396-7 = x, 


a 5 = 15° 55' 


9-43813 


2-69340 


493-6 = x 5 


a 6 = 19° 06' 


9-51484 


2-77011 


589-0 = x 6 




Log. Cos. 


Log p. 


V- 


o = 1S00 - p. 


oj = 3° 11' 


9-99933 


3-25460 


1797-2 


2-8 = o x 


a 2 = 6° 22' 


9-99731 


3-25258 


1788-9 


11-1 = 2 


a 3 = 9° 33' 


9-99394 


3-24921 


1775-0 


25-0 = o 3 


a 4 = 12° 44' 


9-98919 


3-24446 


1755-7 


44-3 = o 4 


a 5 = 15° 55' 


9-98302 


3-23829 


1731-0 


69-0 = o B 


« 6 = 19° 06' 


9-97541 


3-23068 


1700-9 


99-1 = o 6 



The calculations are made by means of logarithms, and the 
figures may be set down as in the above tables. The figures in the 
column headed " Log sin " are obtained from tables of logarithmic 
sines of angles. By adding the value of log It to each of the log 
sines we get the figures in the column headed " Log x " which are 
the logarithms of x x , x 2 , x 3 , &c. The values of x 1} x 2 , x s , &c, are 
obtained by looking up from the tables the numbers which corre- 
spond to these logarithms. 

The calculations for finding the values of the offsets o x , o 2 , o 3 , &c, 
are as given in the lower portion of the table. 

Krohnke's curve tables give values of co-ordinates for setting 
out equidistant points, worked out for a large range of curves. 

Method by Offsets from Chords produced. — In Fig. 238, is the centre 
of a circle of radius R, and AB, BC, CD are successive equal chords 



298 



SURVEYING 



of length I. Let AB be produced to E so that AB = BE. Then 
the triangle CBE is an isosceles triangle and similar to the triangle 

CE CB 

OBC so that we have ~~ = ^5 

CB 2 I 2 
or CE = ?yD~= =5 = length of so-called offset o. 

This geometrical property furnishes us with a method of accu- 
rately setting out a circle. The first chord AB is produced its own 
length to E. On BE as a base a triangle is set out with the tape 

I 2 
having sides BC and CE of lengths equal to I and ^ respectively. 

Point C thus set out is a point on the circle, and BC is a chord of the 

same length as AB. 
By producing BC its 
own length and con- 
structing on it a triangle 
having the same sides 
as before another point 
will be found on the 
curve, and by continued 
application of the 
method any required 
length of curve may be 
set out. 

A tangent to the 
circle at point B would 




^F 



Offsets from Chords produced. 



bisect the length CE so that the perpendicular offset to point C 



from the tangent line would have a length 



2R' 



Therefore, if TA 



is a tangent to the curve the first chord length AB will be set out 

I 2 
by a perpendicular offset GB of length ^5 set off from TA produced. 

Example : A curve of 320 ft. radius is to be set out in 50-ft. 
chord lengths by the chord-offset method. Fig. 239 illustrates in 
detail the setting out of the first two points. 

Offset from chord = i = g^ = 7-81 ft. 

7*81 
Offset from tangent = -=- = 3 -90 ft. 
a 



SETTING OUT CUKVES, ETC. 



299 



To fix point B lay off in the first place a length of 50 ft. from A 
along the tangent line produced and erect an offset of 3-9 ft. This 
will give a point whose distance from A is rather more than 50 ft. 




5i 



T A G 

Fig. 239.— Setting out Curve by Chords and Offsets. 

Adjust the point by moving it slightly parallel to the tangent line 
till the distance AB is 50 ft. and distance BGr 3-9 ft. Point B is 
then accurately fixed. A distance of 50 ft. is then measured from 
point B along the fine AB produced and point E is marked by an 
arrow or otherwise. Then 

to set out point C fix the f 

zero end of the tape at B / / \ 

and hold the mark repre- 
senting 57-81 ft. on the 
tape at point E. An arrow 
is then held inside the tape 
at the 50 mark, which is 
pulled out till both lengths 
are taut. The arrow is 
then at point C on the 
curve. 

Curve Problems. — In Fig. 
240 the two straight lines 
TA and DU are to be con- 
nected by a circular curve 
starting from point A. It is required to find the radius of the 
curve and to set out the curve by offsets from tangent lines. 

If a theodolite is not available the problem may be solved as 
follows : the intersection point B of the lines TA^and DU is first 




Fig. 240.— Curve Problems. 



300 



SURVEYING 



found. Distance BA to the tangent point is then measured and an 
equal distance BD laid off along the line BU. D will be the tangent 
point at the end of the curve. Length AD is measured and the half 
length AH becomes known. 

Then in the right angled triangle AHB we know the lengths AB 
and AH, so that HB can be calculated. HB = ^/ AB 2 — AH' 2 . 
Also if C is the centre of the circle the triangles CAB and AHB are 
CA AH _ . „ AB x AH 



similar, so that -^5 = ^7™ or CA 
Distance BM = 



R = 

R(AB 



HB 
AH) 



and distance AO or OM = 



AH 
R 2 (AB 



AH) 



AB X AH' 

Distances MP and PD are the same as distance AO, and by setting 
ofE the lengths AO and PD to the value found from the above 

formula the tangent line 
I OP will be obtained. The 

/ ; ; \ curve may then be set out 

/ f« \. by offsets from the tan- 

/ / \ gents AO, OP, and PD. 

:' \ The converse of the 

; \ above problem occurs 

when the radius of the 
curve is given. It then 
becomes necessary to find 
the positions of the tan- 
gent points. The inter- 
section point B (Fig. 241) 

is found as before and two 
r u 

arbitrary equal lengths BG 
Fig. 241.-Curve Problems. and BK are measured off 

along the tangent lines. Distance GK is measured so as to find 
the half length GL, and the distance BL is measured or calculated. 
Then A being the starting point of the curve, and C its centre, the 
triangles CAB and GLB are similar so that we have 

AB BL A „ CA x BL R x BL 

CA 




BL ._, CA X 
LG' orAB = — GL 



GL 



By measuring off the length of AB as calculated by the above 



SETTING OUT CURVES, ETC. 301 

formula from the intersection point B along the tangent lines the 
starting and ending points A and D of the curve will be determined 
on the ground. The curve can then be set out by the methods 
already explained. 

Trigonometrical methods may be employed for finding the 
lengths to the tangent points, &c, if the intersection angle is 
measured. In Fig. 240 the radius R is given and the intersection 
angle ABD has been measured and is equal to a so that the angle 

ABC = | 

Then AB = CA cot | = R cot |. 

Also CB = — . But BM = CB - CM = CB - R. 
. a 

8m 2 , 

Therefore BM = — R = ] = 



(tV>) 

\sm x 



sm 2" Vsin ^ \ sm 2 

1 - sin^ x 



a 
cos ^ 

To find the length of the curve AMD. 

Curve subtends angle ft at centre of circle, where ft = 180 — a. 
Length of curve L = R/3 when ft is stated in radians. 
Rft 
~57'30 " 

= H „ „ „ minutes. 

Setting out Curves with Theodolite. — The setting out of circular 
curves with the theodolite depends on the geometrical principle 
that a given length of arc subtends the same angle at any point of 
the circumference. Thus in Fig. 242 if Al, 12, 23 are equal arcs 
the angles 3C2, 3A2, 2A1, 1AB are equal to each other and equal 
to half the angle 302 which the arc subtends at the centre of the circle. 
If I is the length of the arc and R the length of the radius the angle 
subtended at the centre of the circle has been shown to be equal to 



302 
I 



SURVEYING 



R 



X 3,438 minutes. The angle 8 subtended by an arc I at the 

I 

circumference is therefore equal to p X 1,719 minutes. This is 

known as the deflection angle for the curve because a theodolite 
set up at the tangent point A will require to be deflected to the right 
from the tangent line through successive values of the angle in order 
to sight towards the points, 1, 2, 3, &c. The first thing to be done, 
therefore, preparatory to setting out a curve by theodolite is to 
calculate, or find from tables prepared for the purpose, the value of 
the single deflection angle 8, and the values of 28, 38, 48, &c, up to 

n8, where n is the number 
of points to be set out at 
one planting of the instru- 
ment. In practice the curve 
is set out in chord lengths 
instead of arcs, but the 
lengths are made so short 
that the error introduced is 
negligible. Commonly 
accepted limits are, when 
working in chains of 66 ft., 
to set out the points at whole 
chain intervals when the 
curve is over twenty chains 
radius and at half-chain 
intervals when the radius 
is under twenty chains, and 
when working in feet to set out the points 100 or 50 ft. apart according 
as the radius is over or under 2,000 ft. To set out the first point 1 the 
theodolite is planted at A and sighted along the tangent line in 
the direction AB with the vernier set to zero. The single deflection 
angle 8 is then turned off to the right so that the theodolite points 
in the direction Al. At the same time one end of the chain is held 
at A, an arrow is held on the chain at length I from A (this is usually 
at the other end of the chain), and the chain is straightened out and 
swung about A as a centre until the arrow is brought exactly into 
the line of sight of the theodolite. The arrow is then at point 1 
on the curve and may be fixed in the ground, or the point may be 
marked permanently by driving in a peg. To set out point 2 the 




Fig. 242.— Setting out Curve by 
Theodolite. 



SETTING OUT CUEVES, ETC. 303 

theodolite is further turned to the right through a deflection angle 
so that the reading on the circle becomes equal to 28. It then 
looks towards point 2. The end of the chain is now held at point 
1 and a length I of the chain is taken and swung about this point till 
the arrow is in the line of sight of the theodolite. The arrow then 
marks point 2. Further points are set out in succession round the 
curve by turning the theodolite through a deflection angle each time 
and measuring a distance I from the point last fixed. 

When in proceeding with the setting out the points become 
inconveniently far away from the instrument or when the view 
becomes obstructed, and in any case before the sum of the deflection 



Fig. 243. — Changing Position of Theodolite. 

angles reaches 45°, the theodolite must be shifted forward to the 
point last set out and a fresh start made from a new tangent line, 
as shown in Fig. 243. The curve has been set out, let us suppose, 
up to point 4 with the theodolite planted at point A and the con- 
ditions render it necessary that the theodolite should be shifted 
forward in order to continue the setting out. It is planted at 4, 
the vernier is set back to zero, the telescope is sighted back on 
point A and the lower plate is clamped. The telescope is then 
transited so as to point forward towards C along the line A4 pro- 
duced. The vernier reading is still zero. The telescope is now 
turned to the right through an angle equal to the angle used in 
setting out point 4, that is in this case through an angle equal 
to 48, and points towards D. An inspection of the figure will show 



304 SURVEYING 

that the telescope is now looking along the line of the tangent at 
point 4. Point 5 will, therefore, be set out by turning the telescope 
further to the right through the deflection angle 8 making the reading 
on the circle equal to 58 or the same as it would have been if 
point 5 had been set out from A. Points 6, 7, &c, are set out by 
turning the telescope so that the circle readings are equal to 68, 
78, &c 

Instead of setting the vernier to read zero when sighting back 
from the turning peg (peg 4) to the tangent point, it is often more 
convenient to set it to read an angle to the left of zero equal to the 
whole deflection angle of the turning point. In this case we would 
sight on A with the vernier reading the angle 360° — i8, so that when 
the telescope is transited and turned to the right through the angle 
48 the reading will be zero and the telescope will be pointing along 
the tangent. Points 5, 6, 7, &c, will then be lined in by again setting 
off the deflection angles 8, 28, 38, &c, to the right from point 4, 
exactly as from point A. 

Example. The centre line of a portion of railway is being set 
out and marked by pegs driven in at every chain length of 100 ft. 
A curve of 2,500 ft. radius occurs connecting two straights. We 
will follow through the operations and calculations required in 
setting out the curve. 

The first operation is to find the intersection point of the straight 
lines. The lines are ranged out roughly by eye in the first place so 
as to find the approximate position of the point. Then with the 
theodolite planted on one of the straights and looking along the line 
towards the intersection point two arrows are ranged into line a few 
yards apart and so as to occur one on either side of the required 
point. A string is stretched between the arrows. The theodolite 
is then planted on the other straight, the telescope is sighted into 
line and pointed towards the intersection point. An arrow is then 
set at the point where the line of sight cuts the string and marks the 
intersection of the straights. A peg may be driven as a more 
permanent mark. 

The theodolite is now set up over the intersection point and the 
angle between the straights is measured. Assume that this is 
148° 20'. The distance from intersection point to tangent point 
can now be calculated and must be found to enable the tangent 
points to be set out. 



'alf intersection angle 


= 74° 10'. 


istance to tangent point 


= R cot - 
A 


Log 2,500 
Log cot 74° 10' 


= 3-39794 
= 9-45271 



SETTING OUT CURVES, ETC. 305 



2,500 cot 74° 10'. 



2-85065 = log 709-0. 

The distance of 709 ft. is, therefore, measured backwards along 
each straight from the intersection point in order to fix the starting 
and finishing points of the curve. 

On pegging out the straight up to the commencing point of the 
curve it will be a mere coincidence if a peg occurs at the tangent 
point. Let us assume that the distance from the last peg on the 
straight, say, peg No. 71, to the tangent point measures 39 ft., so 

.-70 9 :;"-"-""" %s$"""~~-~.. 

^~~7S~~~7S 77 78 7S~~sT 




Fig. 244. — Setting out Curve. 

that the distance from the tangent point to the first peg on the 
curve, peg No. 72, will be 61 ft. 

Deflection angle for 61 ft. = k-^k X 1719 = 42 minutes. 
b 2,500 

Deflection angle for 100 ft. = ^^ X 1719 = 68"76 minutes. 
& 2,500 

Angle which arc of 100 ft. subtends at centre of circle = 68-76 
X 2 = 137-52 minutes. 

This latter figure may be used in finding the whole length of the 
curve. The angle which the whole curve subtends at the centre of 
the circle is equal to 180° minus the intersection angle. In this 
case the curve subtends an angle of 180° — 148° 20' = 31° 40' = 

1,900 minutes. Then length of curve = 100 X -^L = 1381-6 ft. 
6 137-52 

The distance from starting point to first peg on curve is 61 ft., 

leaving a remainder of 1320-6 ft. Therefore, following on the first 

peg 13 complete arcs of 100 ft. will require to be set out, and there 

s. x 



306 SURVEYING 

will be a closing length of 20-6 ft. to the end of the curve. See 
Fig. 244, which shows the curve with the positions of the pegs 
plotted to scale. 

The following is a list of the angles required in setting out the 
points. The angle to set out the first point, peg No. 72, is the 
deflection angle for 61 ft., namely, forty -two minutes. It is very 
desirable to get rid of the occurrence of this odd amount in the 
series of deflection angles, and this is effected by commencing with 
the theodolite set to read forty-two minutes to the left of zero, 
that is, the angle 359° 18', so that when the theodolite is turned 
from the tangent to the right through 42', it will be reading 0° 0' 
and pointing to peg 72. The deflection angles for the first ten pegs 
for a wide range of curves can be obtained without calculation from 
various sets of curve tables, such as Krohnke's. 

Angles for Setting out Curve of 2,500 ft. Radius in 
Arcs of 100 ft. 



Tangent 


359° 18' 


Peg No 


77 


5° 43f ' 


Peg No. 72 


0° 0' 




78 


6° 521' 


73 


1° 8f 


,, 


79 


8° Oil' 


„ 74 


2° 17*' 




80 


9° 10' 


75 


3° 26|' 


,, 


81 


10° 18|' 


» 76 


4° 35' 


" 


82 


11° 27*' 



The theodolite being planted at the start of the curve and sighted 
along the tangent line with the vernier set to read 359° 18', peg 
No. 72 is set out by turning the theodolite to the right through 
42' so as to read 0° 0' and lining in the peg at a distance of 61 ft. 
from the tangent point. The succeeding pegs are set out in turn by 
the deflection angles shown in the table and at distances of 100 ft. 
from each other. Even although the view is unobstructed right 
round the curve it would be advisable to shift the theodolite once, 
say, after peg No. 79 has been fixed. Having removed the instru- 
ment and planted over peg No. 79 set the vernier to read to the left 
of zero an amount equal to the whole deflection angle already 
turned through. In this case the whole deflection angle up to 
peg 79 is 8° 01£' + 0° 42' = 8° 43J', and we set the vernier to read 
360°— 8° 43|' = 351° 16| ' and sight back to the tangent point. 



SETTING OUT CURVES, ETC. 307 

The telescope is then transited and turned to the right to read 
0° 0' when it points along the tangent at peg 79. The succeeding 
pegs will then be lined in by again laying off the same series of 
deflection angles, that is, peg 80 is given by the angle 1° 8f ', peg 81 
by the angle 2° 17f , and so on. The accuracy of the work is tested 
at the closing length from peg No. 85 to the end tangent point. 
This distance should measure the calculated amount, namely, 
20*6 ft., and the angle read when the theodolite is sighted on to the 
tangent point, added to the total deflection angle up to peg 79, 
should be equal to half the angle subtended by the curve at its 
centre, in this case half of 31° 40' or 15° 50'. 

Curve to Left. — The foregoing descriptions of the setting out of 
curves by theodolite apply to curves which turn off to the right hand 
from the tangent line. A slight difference in the procedure arises 
in setting out a curve to the left hand due to the fact that theodo- 
lites in this country are usually graduated only in right-hand or 
clockwise direction. If, therefore, in setting out a curve to the left 
we start with the telescope looking along the tangent line and the 
circle reading at zero, the angle for setting out the first point will be 
got by subtracting the deflection angle from 360°. The angle for 
the second point will be obtained by subtracting the deflection 
angle from the angle for the first point and so on. The subtraction 
of degrees, minutes, and fractions is an awkward process and liable 
to be a fruitful source of arithmetical error, and should, therefore, 
be avoided if possible. This may be done by proceeding as shown 
in Fig. 245 and accompanying table, which give a comparison of 
the angles required for setting out the same curve to the right 
hand and to the left hand. The angles are calculated for a sufficient 
number of points as for a curve to the right. Then to set out a left- 
hand curve these angles are taken in the reverse order. In the case 
illustrated the theodolite is sighted along the tangent line with the 
circle reading set to 9° 35'. Then peg No. 1 to the left is set out by 
the angle 8° 37|', peg No. 2 by the angle 7° 40', and so on. 

Inaccessible Intersection Point. — Referring to Fig. 241, if the 
intersection point B is inaccessible it may be possible to run and 
measure some line such as GK between the tangent lines and to 
measure the angles AGK and GKD. The lengths of the sides of 

x2 



308 



SURVEYING 



Angles 


for Curve to Left. 


Angles for Curve to Right. 


Tangent 9° 35' 
Peg No. 1 = 8° 37$' 
2 = 7° 40' 


Tangent 0° 0' 
Peg No. 1 = 0° 57-1' 
2=1° 55' 




} 


3 = 6° 42-1' 




3 = 2° 521' 






4=5° 45' 




4 = 3° 50' 






5 = 4° 47$' 

6 = 3° 50' 




5 = 4° 47$' 

6 = 5° 45' 




> 

5 


7 = 2° 52$' 
8=1° 55' 
9 = 0° 57$' 
10 = 0° 0' 




7 = 6° 421' 
8=7° 40' 
9 = 8° 37$' 
, 10 = 9° 35' 




Fig. 245. — Curves to Eight and Left, 
the triangle BGK and the tangent lengths BA and BD could then 
be calculated. 

Angle BGK = 180° — AGK 
„ BKG = 180° — GKD 
and intersection angle GBK = 180° — (BGK + BKG). 

In the triangle BGK the base GK and the angles at G and K are, 
therefore, known, and the lengths BG and BK can be calculated by 



SETTING OUT CURVES, ETC. 309 

the method given on p. 188. Also, since the intersection angle at 
B is known, the tangent lengths BA and BD can be calculated by 
the method given on p. 301. The distances GA and KD are then 
found by subtraction, and when measured off backwards from 
points G and K serve to fix the tangent points on the ground. The 
setting out of the curve can then proceed in the usual manner. 

Obstructions in Setting out Curves. — Any point on a curve can 
be set out independently of any other by laying off its whole deflec- 
tion angle and measuring the chord length along the line so given. 
If the whole deflection angle is A the chord length is = 2 R sin A, 
R being the radius of the curve. This points to a method of passing 
obstructions which occur on the line of the curve, or on the lines 
of sight of the theodolite. The setting out of the points affected 
by the obstruction is omitted and the next point beyond the obstruc- 
tion is set out by the whole deflection angle and chord length. 
The curve is then continued in the usual manner, and the omitted 
portion may be dealt with when the obstacle is removed. 

Many of the obstacles in the nature of obstructions to the line of 
view of the theodolite can be easily overcome by suitably choosing 
the points at which to shift the instrument. 

Where the line of the curve is much obstructed the difficulty may 
often be overcome by running a parallel curve of smaller or larger 
radius. The actual curve is then set out by radial offsets as the 
obstructions are removed. If it is desired, in the case of a parallel 
curve, to keep the correct centre line chainage the interval between 
the points must be altered in proportion to the increase or reduction 
of the radius, but the deflection angles will remain the same. The 
setting out of a parallel curve has the advantage that the pegs may 
remain and be of use during construction of the works, whereas 
centre-line pegs are ordinarily lost as soon as construction com- 
mences. 

Setting out Building Work. — The principal operations required 
in the setting out of building work are the laying off on the ground 
of right angles and other angles with the theodolite or otherwise ; 
the setting out of straight and curved lines ; and the accurate 
measuring and marking-off of distances. Usually only the prin- 
cipal building lines are required to be set out by the engineer or 
architect. Wooden pegs or stakes are driven into the ground to 



310 SURVEYING 

mark the lines and points, accurate points being denned by nails 
driven into the heads of the pegs. 

Methods of setting out angles, ranging lines, &c, with and 
without the theodolite, are given in Chapters VII. and XV. In the 
measuring of distances much greater precision is required than in 
ordinary surveying. In important work, such as foundations to 
carry a steelwork superstructure, the essential dimensions should 
be set out accurately to within J in. On rough or sloping ground 
and in transferring lines, from the surface of the ground to the 
bottom of foundation trenches, &c, very careful use of the plumb- 
bob is necessary if accurate work is to be accomplished. 

An example of the setting out of the principal lines of building 
work is illustrated in Fig. 246, which shows in plan and elevation 
a bridge carrying a road over a railway. The principal building 
lines are the face lines of abutments and wing walls at the level of 
the top of the concrete foundation, and it is these which would 
ordinarily be pegged out. Before the bridge can be set out the 
engineer must know the chainage to the centre of the bridge 
(point A), the angle of skew, the clear span and clear width between 
parapets, and must have a drawing showing the masonry in plan. 
Dimensions of wing walls, &c, may be marked on the drawing, or 
may require to be scaled. 

The skew distances AE, ME and EK will not usually be marked 
on the drawing, but are required in the setting out. They might 
be scaled from an accurate drawing, but should be calculated. The 
corresponding square distances are 13 ft. 6 ins., 17 ft. 6 ins., and 
19 ft. 6 ins. The skew distance in each case is obtained by multi- 
plying the square distance by the cosecant of the angle of skew 
(65° 10'), and the results are 14 ft. 10| ins., 19 ft. 3| ins., and 
21 ft. 5| ins. respectively. The first operation in the setting out is 
to put in the centre peg at A. On a straight centre line of railway 
already staked out the theodolite would be set up on an adjacent 
centre-line peg, sighted on to another centre-line peg at some 
distance away, and then tilted down to line in the peg at A, whose 
position is fixed by measurement from the nearest centre-line peg. 
The exact point is marked by a nail. The theodolite is then set up 
over peg A and centred over the nail. With the vernier set to zero 
the telescope is sighted on a centre-line peg B at some distance 
away, and an angle of 65° 10' is then turned off to the right. The 



SETTING OUT CURVES, ETC. 

Surface of c JyQkJjUHlcj , Roadway 



311 



Surface of 




Ground 



Elevation. 




Plan. 

Fig. 246.— Setting out a Bridge. 



312 SURVEYING 

telescope now gives the centre line of the road and peg E is lined 
in at 14 ft. 10| ins. from A. Transfer peg G is also lined in clear 
of the site of the excavation, and with the telescope transited cor- 
responding pegs F and H are put in on the other side of A. The 
theodolite is then set up over E and sighted on a mark C set out at 
13 ft. 6 ins. from B square to the centre line of railway. This 
gives a line of sight parallel to the centre line of railway, and points 
K and L may be lined in at the distances marked, and, by tran- 
siting the telescope, points M and N may also be lined in. The 
corners of the wing walls are fixed by square offsets of 8 ft. from 
pegs N and L. The face line of the other abutment would be set 
out in a similar manner, with the theodolite set over peg F. When 
the building-line pegs 0, M, E, K, &c, have been put in, the outline 
of the concrete foundation may be marked out on the ground in 
accordance with figured or scaled dimensions obtained from the 
drawing, working from the building line as a base fine. When 
excavation commences, however, these pegs will be lost, and it is 
necessary to have such transfer pegs as will enable the building lines 
to be recovered at any time, and particularly when building is 
about to commence on top of the concrete foundation. A line 
stretched over pegs G, A and H will give the centre fine of the road 
at any time, and points E and F on that fine can be recovered by 
measuring the proper distance from A, and can be transferred to 
the surface of the concrete by plumbing. A fine stretched from 
L to N will give the face fine of the abutment, and points M and K 
can be obtained by measurement from E, as in setting out. It 
would be advisable, however, to have transfer pegs for each wing 
wall, such as P and Q, set out at a round number of feet from the 
corners and so as to be clear of the excavation. These preserve the 
line of the wing wall and enable points and M to be readily 
recovered and transferred down to the foundations. 



CHAPTER XX 

CALCULATION OF AREAS 

In this chapter consideration is given to the methods in common 
use for computing areas from a survey plan, such as by dividing the 
area into triangles, parallel strips, squares, &c, also the determina- 
tion of areas from offsets by the trapezoidal and Simpson's rules, 
and the finding of areas by the planimeter. Methods of computing 
areas of plots and areas of traverses directly from the field measure- 
ments are dealt with, and the effect of shrinkage of plans and the 
allowance to be made for shrinkage in taking out areas are 
considered. 

Square Measure. — The British units of area employed in sur- 
veying are as given in the following statement : — 

1 sq. ft. — 144 sq. ins. 
1 sq. yard = 9 sq. ft. = 1,296 sq. ins. 
1 sq. pole = 30| sq. yards = 272£ sq. ft. 
1 sq. chain = 484 sq. yards. 
1 rood = 40 sq. poles = 1,210 sq. yards. 
1 acre = 4 roods = 10 sq. chains = 4,840 sq. yards = 43,560 
sq. ft. 

The following figures give the relationships existing between 
some of the metrical and some of the British units of area : — 

1 sq. metre = 10-7639 sq. ft. = 1-1960 sq. yards. 

1 hectare = 10,000 sq. metres = 2-4710 acres. 

1 sq. kilometer = 100 hectares = 0-3861 sq. miles. 

1 sq. ft. = 0-0929 sq. metres. 

1 sq. yard = 0-8361 sq. metres. 

1 acre = 0-4047 hectares. 

For small areas, such as building lots, the unit of area commonly 
used is the square yard, occasionally the square foot. For large 
areas the acre is almost universally employed. The method of 



314 



SURVEYING 



expressing fractions of an acre in roods, poles, &c, is very cumber- 
some. It is much simpler to express the fraction decimally. 

The following table gives formulae for the areas of the geometrical 
figures which are most commonly of use in the calculation of survey 
areas : — 

Areas of Geometrical Figures. 



Figure. 



Triangl 




Area = ±bh. 





Area = 

Vs(s —a)(s — b) (s — c) 

, a + b + c 
where s = = 



Parallelograms. 



PIP 

L — / — J L — ,- j 



Area = Ih. 



Quadrilateral. 





Area = i&(A x + //,) 
or = IBH. 



CALCULATION OF AREAS 



315 



Trapezium. 

r — s* — i 



Area= |A(sj. + s 2 ). 



Circle. 




Area = 3-1416 r 2 

or = 0-7854 d 2 . 



Sector of Circle. 




Area = ^ Ir 

3-1416 r 2 

or= ^60 

= 0-008727 r 2 0, 
being given in degrees. 



Segment of Circle. 




V 



r 2 (0-0087270° -£ sin 0), 
being given in degrees. 



Flat Segment of Circle. 



Parabola. 




b w_ 



Area = y Ih approx. 
o 



Area = ~bh. 
o 



316 SURVEYING 



Figure. 


Area. 


Ellipse. 


Area = 0-7854 ab. 



Area of Land. — For surveying purposes the distance between 
two points is taken as the straight distance between the projections 
of the points on a horizontal plane. Similarly, the area of a plot 
of land is not taken as the whole surface exposed following the 
undulations and irregularities, but is taken as the area contained 
within a projection of the boundaries on a horizontal plane. Areas 
measured from an accurate survey plan will fulfil this condition. 

Methods of taking out Areas. — There are two general methods of 
obtaining areas : — 

(a) By scaling or measuring from a survey plan. 

(b) By direct calculation from field measurements. 

Areas are most commonly obtained by scaling from a plan. 
Greater accuracy can be obtained by method (6), but this method 
is in general only applicable when the measurements have been 
specially taken and arranged with the view to computation of areas. 

Areas from Survey Plan. — (1) By dividing the area up into 
geometrical figures. The plot whose area is required may be 
divided up into geometrical figures of the forms shown in the fore- 
going table. Its area will be obtained by adding together the areas 
of the separate figures. 

Any straight-sided figure may conveniently be divided up into 
triangles. The polygon ABCDEF (Fig. 247) may, in order to calcu- 
late the area, be divided into the four triangles shown. The 
triangles are taken in pairs. AC is scaled as the common base of 
the triangles ABC and AFC, and FD is also scaled as the common 
base of two triangles. The heights h v h 2 , h 3 , and h^ are also scaled 
off. The area of the figure is then equal to \ AC (h x + h 2 ) -f | FD 
(h+h). 



CALCULATION OF AREAS 



317 



Any straight-sided figure may, by simple geometrical construction, 
be reduced to a single triangle of equal area, and this furnishes 
a useful and rapid method of taking out areas. It is required 
to construct a single 
triangle equal in area 
to the quadrilateral 
ABCD (Fig. 248). 
Divide the figure up 
into two triangles by 
the diagonal BD. 
Through C draw a 
parallel to BD to meet 
AD produced in E. 
Then if line BE be 
drawn in, the triang 
BCD and BED will be of equal area since they have the same base 
BD and their perpendicular heights from base to apex are the same. 
The area of the quadrilateral ABCD formed of the two triangles 
ABD and DBC is, therefore, equal to the area of the triangle 
ABE formed of the two triangles ABD and BED. By continued 
application of the above method any figure having a number of 




, Fio. 247. — Area of Figure by Subdivision. 



straight sides may be reduced to a 



single triangle. The con- 
structions to obtain a single 
triangle equivalent in area to 
the six-sided figure ABCDEF 
are illustrated in Fig. 249. 
Eg is drawn parallel to DF. 
This gives triangle D#F 
equivalent to triangle DEF. 
Bh is drawn parallel to CA, 
giving triangle ChA equiva- 
lent to triangle CBA. 
Finally, Ck is drawn parallel 
to Dh, giving triangle DM 
equal to the triangle DCA. 
The whole triangle kDg is then equal in area to the figure ABCDEF. 
The foregoing method of reducing a figure to a single triangle is 
rapid, as only a few of the construction fines shown on the diagram, 
or portions of them, need actually be drawn. Thus to obtain 




Fig. 248.— Single Triangle equal to 
Quadrilateral. 



318 



SURVEYING 



point g the edge of the parallel ruler (or set-square) would be laid 
across the points F and D and the ruler would then be rolled till the 
edge passed through E. A short stroke drawn across the line AF 
produced would then fix point g. 

The degree of accuracy with which the area of the single triangle 
corresponds to the area of the figure will depend on the draughts- 
manship. The benefit of the method lies in the greatly reduced 
amount of scaling and calculation involved in finding the area of a 
single triangle as compared with several separate triangles, and the 
correspondingly reduced liability to error. The method may often 
be employed to give an independent check on an area calculated in 

some other way, and 
D vice versa. 

C ■ sy%. Methods of dealing 

with irregular and 
curved boundaries are 
illustrated in Fig. 250. 
Irregular boundaries 
are " equalised " by 
replacing them with 
straight sides arranged 
so as to include and 
exclude equal small 
portions of the area as 
nearly as may be judged by the eye, the purpose being to obtain 
a straight -sided figure of exactly the same area as that contained 
within the irregular boundary. The straight-sided figure would 
be dealt with by the method of triangles. 

The irregular sides AB and ED (Fig. 250) are equalised by the 
dotted lines shown. 

Regular curved boundaries, such as BCD in the figure, are 
generally dealt with by dividing up into flat segments, such as BC 
and CD, the area of these being calculated by the formula for flat 
segments. Thus the whole area of the above figure would be 
equal to 

i BE (h, + h 2 ) + i BD h 3 + f BC h, + I CD h & . 

(2) By dividing the area up into parallel strips of equal width. 
The usual method of doing this is to draw parallel lines en a piece of 




Fig. 249.— Area by Single Equivalent Triangl 



CALCULATION OF AKEAS 



319 



tracing cloth or paper and to lay this over the area, thus dividing 
it into strips. The area contained in any strip will be equal to the 
mean length between boundaries multiplied by the width of the 
strip, and the whole area will be equal to the sum of the lengths of 
all strips multiplied by the width of a strip. The width of strip 
should be such as will render the calculation of the area easy. If 
the area is desired in actual square inches of paper the width of 
strip would be made either 1 in. or some convenient simple fraction 
or multiple of an inch. If the area is to be taken from a plan and 
given in acres the width of strip would be taken as equal to one 
chain to the scale of the plan or some suitable fraction or multiple 




■within Irregular Boundaries. 



of a chain, because of the simple relationship which exists between 
square chains and acres, namely one acre = ten square chains. 
The width of strip should also be limited to such a size as will cause 
the portions of boundary intercepted between adjacent lines to be 
sensibly straight, so that the mean length of a strip may be taken as 
the length along its middle line. 

The method is illustrated in Fig. 251, where ABCD is the figure 
whose area has to be determined, and the dotted fines represent the 
parallel lines drawn on a superposed piece of tracing cloth or paper. 
The whole area with the exception of an irregular portion at each 
end is comprised within parallel strips. The areas of these end 
portions require to be separately calculated. The area of the 
remainder is equal to d (l ± -f- l 2 + 1- l n ), where d is the width of 



320 



SURVEYING 



strip and l v l 2 , &c, are the mean lengths of successive strips. There 
are several devices for rapidly summing up the lengths of strips. 
Where the lengths are short they may be summed up with a pair 
of dividers, the points being set first to the length of the first strip, 
and then further opened out by the length of the second strip, and 
so on. Another simple method is to take a long strip of paper and 
mark off the lengths successively along this. The total length 
obtained may be scaled in one operation. An improvement on the 
above method consists in summing up the lengths by means of a 

scale with a sliding index. 
The graduations of the scale 
may be arranged so that the 
reading of the index gives the 
area in acres. The appliance 
is then known as a " comput- 
ing scale." 

(3) By dividing the area up 
into squares. This method 
is not essentially different from 
the preceding. Parallel lines 
are drawn on tracing cloth or 
paper in two directions at 
right angles to each other so 
as to form squares of a con- 
venient fraction or multiple 
of the unit area employed. 
The tracing is laid down and 
fixed over the figure whose 
area is to be ascertained. The area comprised within the boundary 
of the figure will consist of a certain number of whole squares 
together with a number of fractions of squares of triangular and 
trapezoidal form around the boundary. The number of whole 
squares is counted, and the areas of the separate fractions are 
separately calculated. 

The method of finding an area by dividing it into squares is 
generally less convenient than the method by parallel strips. 

(4) Areas by means of offsets. The method of taking out areas 
by offsets is most commonly applicable where the area is in the 
form of a narrow strip which continues in one general direction for 






t 

J- /, -A 

""x:;;:;;:^":::;:::j 
V d 





,. — i n — . 






a 



Fig. 251. — Area by Parallel Strips. 



CALCULATION OF AREAS 



321 



some distance. In dealing with plan areas the offset method may 
be adopted for irregular boundaries instead of the equalising method 
already explained. The external sides of the triangles or other 
shapes into which the figure is divided are then kept entirely 
within the boundary, and the excluded portions of area are dealt 
with by the offset method. The offset method is also adapted to 
finding areas directly from measurements taken on the ground. 

The offsets may be taken at equal or unequal intervals apart. 
The usual case of offsets taken at equal intervals from a straight 
base line to a curving boundary is illustrated in Figs. 252 and 253. 

The common distance between offsets is d, the lengths of the end 





=S 


*! 








* 


* 


•S 


,-- d~i 


<- d — * 











Fig. 252. 



r — __ 














£ 


k 


% 


* 


* 










*-- d~+ 


<-■ d ~* 











Fig. 253. 

Areas by Offsets. 

offsets are a and b, and the lengths of the intermediate offsets are 
successively y x , y 2 , y 3 , &c. If the boundary is straight between 



successive offsets the total area will be equal to d[ ~ -f- y x -f y 2 + y 3 
+ , &c, + n ) or otherwise equal to d ( — « — -{-2y ) (trapezoidal rule), 

where a and b are the lengths of the end offsets or ordinates and 

%y is the sum of all the intermediate ordinates. 
A slight variation of the above method consists in measuring 

the offsets at the middle of each equal space, as shown in Figs. 254 

and 255, instead of at the ends of the spaces. The area is then 

equal to d2,y (mean ordinate rule). 
Where the boundaries are regularly curving the area will be more 
s. Y 



322 



SURVEYING 



accurately determined by means of Simpson's rule than by either 
of the two foregoing rules. To apply Simpson's rule the ordinates 
are taken as in Figs. 252 and 253, and there must be an even number 

of equal spaces. Referring 
to these figures the rule 
is : — 

d 

Area = ^ (a + Ay x + 2y 2 

+ 4 ;¥3 + 2t/ 4 +4i/ 5 +6). 

Where, as in Fig. 256, 

the boundary is a series 

of straights of varying 

lengths, or where it can 

be equalised into such a 

form, the obviously correct 

method of taking out the 
Areas by Mean Ordinates. . , „ , , , 

J area is by onsets taken 

to the angles of the boundary. The whole area will be obtained by 

adding together the areas of the separate trapezoids into which 

it is divided by the offsets. Thus the area of Fig. 256 is equal to 

K (Vi + yz) + ¥2 (y 2 + y 8 ) + K iy* + yd + K (& + y^ 

Stated in a more convenient form for calculation, the area is equal 
to \{ 1 Jih + 2/2 s 2 + 2/3S3 + 2/4S4 + y & s 6 ), where s 2 is the sum of the two 
spaces adjacent to 
the ordinate y 2 , and 
so on, as shown in 
the figure. In tak- 





Fig. 254. 






c 


S 




N h 


1 " 1 < 


' j, ' 


/ 



Base Line 
Fig. 255. 



a survey plan by 
this method, the 
distances s x , s 2 , s 5 , 
&c, would be scaled 
off complete. They 







J *3 




% 




% 






Base 


Line 








... 


— d, 


4 d 2 *"-df- 


+-- d t 


*j 


<--- 


—S, 


-*. s 





H 


< s 5 


i 



St 

Fig. 256 



Area by Offsets. 

would not be economically obtained by scaling the separate spaces 
d v d 2 , d 3 , &c, and adding them together. 

(5) Areas by Planimeter. There are various forms of planimeter 
or mechanical instrument for the measurement of areas. In most 
forms the operation of finding an area consists in guiding the tracing 



CALCULATION OF AREAS 323 

point of the instrument once completely round the boundary of the 
figure and then taking the reading of an index on a scale. The 
areas of figures of irregular outline are much more easily obtained 
by planimeter than by any other means. 

Amsler's polar planimeter will be described, as it is in very com- 
mon use for measuring areas. The essential elements of the instru- 
ment are shown diagrammatically in Figs. 257 and 258, which 
illustrate the two forms in which it is constructed. In each case 
there are two bars hinged together. One bar is fitted with a tracing 
point at one end and has a freely revolving roller with a graduated 
scale reading against a fixed index near the other end. The other 
bar has at its extremity a sharp point for fixing into the paper. 
In the form shown in Fig. 257 the tracing point D and the roller B 
are on opposite sides of the hinge C, while in the form of Fig. 258 




Fig. 258. 
Elements of Planimeter, 

the tracing point and the roller are on the same side of the hinge. 
The instrument is fixed to the paper in each case at the point A 
and rests on the roller B and tracing point D, which slide over the 
paper during the operation of measuring an area. 

In using the planimeter the tracing point is set to a definite 
mark on the boundary, the roller scale is set to zero, and the tracing 
point is then moved in a clockwise direction round the boundary 
and back to the starting point. The whole instrument swings 
meantime about the point A and the motion of point B will be 
partly by sliding and partly by rolling. While the tracing point is 
making the circuit the roller will revolve a certain amount forwards 
and a certain amount backwards, but for a complete circuit of an 
area there will always be a balance of rotation in one direction. 
The amount of this rotation is proportional to the area of the figure, 
and is read off on the scale. 

y2 



324 



SURVEYING 



The actual construction of two forms of Amsler's planimeter 
is shown in Figs. 259 and 260. In Fig. 260 the hinge is between 
the roller and the tracing point and the hinge and roller are at a 
fixed distance apart, being attached to a frame which slides on the 
bar carrying the tracing point. The sliding frame carries also the 
vernier and counter for recording the rotation of the roller. Areas 
can be read off directly in the required units to various definite 
scales by altering the distance from the tracing point to the hinge. 
The required distances for the various scales are shown by marks 



Pig. 259 




Fig. 260. 



Amsler Planimeters. 



engraved on the bar. There is usually a tangent screw for the 
accurate adjustment of the sliding frame to these marks. 

The form of instrument in which the roller is between the hinge 
and tracing point is shown in Fig. 259. Here again the roller and 
hinge are at a fixed distance apart, while the distance from the 
tracing point to the roller can be altered. The centres of tracing 
point, roller and hinge lie in one straight line, although the bars are 
crooked in various ways to comply with the requirements of folding, 
housing of the roller, &c. 

To get accurate results with the planimeter the surface on which 



CALCULATION OF AREAS 325 

it works must be smooth and level. In the usual case of a small 
area the fixed point is set in the paper at some distance outside the 
boundary of the area and a trial run of the tracing point is made 
round the boundary to see that every part is reached. The tracing 
point is then set to the starting mark and slightly pressed into the 
paper, the instrument is gently raised off the paper at the roller 
and the roller is turned by finger until the index reads zero. It is 
then carefully lowered on to the paper again. The tracing point is 
then guided once round the boundary in a clockwise direction and 
brought exactly back to the starting point. The index reading 
then gives the area. As a check on the work, allow the first index 
reading to remain unaltered and run the tracing point round a 
second time. The reading now obtained should be double the first 
reading and equal to twice the area. A large discrepancy would 
indicate that a mistake had occurred. 

Large areas are dealt with by setting the fixed point within the 
boundary. Then as the tracing point completes the circuit of the 
boundary the whole instrument swings through a complete revolu- 
tion about the fixed point. The area is obtained by adding a 
constant to the index reading. Values of the constant for various 
scales are engraved on one of the bars. 

Areas by Direct Calculation from Field Measurements. — The 

measurements required for the calculation of the areas of enclosures 
of various forms are indicated on the figures on pages 314 to 316. 
For a triangular field the lengths of the three sides may be measured, 
but the formula for the area in that case does not lend itself to easy 
calculation. The more usual method for a triangle is to take one 
of the sides as a base and with the optical square set out a perpen- 
dicular from it to the opposite corner. The lengths of the base line 
and the perpendicular offset from it to the opposite corner are 
measured and half their product gives the area. 

To measure the area of a quadrilateral one of the diagonals may 
be set out and measured and from it offsets taken to the two opposite 
corners. The areas of the two separate triangles into which the 
figure is divided may then be calculated and the whole area obtained. 
The areas of polygonal figures may also be obtained by dividing them 
up into triangles and taking sufficient measurements to enable the 
area of each triangle to be calculated. 



326 



SURVEYING 



It may sometimes be more convenient in the case of a quadri- 
lateral to measure a long side as a base line, as shown in Figs. 261 
and 262, instead of running in a diagonal. Accurately squared 
offsets are set out and measured to the two corners opposite the 
base, and the positions at which they occur on the base line are 

accurately mea- 
sured. Referring 
to the figures the 
area in each case 
is equal to \ (y 1 
XD + y 2 CB). 

A method of 
dealing with an 
enclosure having 
an irregular side 
is shown in Fig. 
263. A base line 
EF is set out in 
a convenient posi- 
tion close to the 
irregular bound- 
ary, thus dividing 
the figure into a 
quadrilateral 
EFCD and a strip 
ABFE. The area 
of the former may 
be obtained by 
either of the 
methods already 
described. To 
obtain the area of 
the strip offsets are set out from the base line to each change of 
direction of the boundary and their lengths and positions on the 
base line are measured. Methods of calculating the area from the 
offsets have been described in the preceding pages. 




Fig. 263. 
Areas from Field Measurements. 



Area of a Traverse. — The area contained within the survey lines of 
a closed traverse may readily be obtained when the co-ordinates of 



CALCULATION OF AREAS 



327 



the stations have been calculated, and if the survey lines have been 
used to locate the boundaries of an enclosure by means of offsets 
then the whole area of the 
enclosure can be found by 
calculation from the 
measurements alone. , The 
method of finding the area 
of a closed traverse is illus- 
trated in Fig. 264. The 
stations are numbered in 
consecutive order one way 
round and the co-ordinates 

x i> Vi> x 2> y<t> &c -> are num- 
bered to correspond with the 
stations. The area of the 
simple traverse shown will 
evidently be obtained by measuring the area of the figure A123D 
and subtracting from it the area of A143D or in detail 

Area of 1234 = A12B + B23D — A14C — C43D 

= \{x % — x x ) {y x + y 2 ) + i (x 3 — x 2 ) (y 2 + y 3 ) 
— |(x 4 — x x ) (y 1 + y±) — \ {x 3 — z 4 ) (y 4 + y 3 ) 

This works out to 

2 [yi( x 2 — x ±) + y*( x 3 — x i) + 2/3K — x %) + 2/4(^1 — ^ 3 )]- 

The general rule may be expressed in words as follows : Multiply 
the ^/-ordinate of each station by a;-ordinate-of-the-f ollowing-station- 
minus-z-ordinate-of-preceding-station. Half the sum of these 
products gives the area. 




Correction for Shrinkage of Plan. — Plans drawn on paper are liable 
to undergo slight alterations in dimensions, which may be contrac- 
tions or expansions, due principally to atmospheric changes. If 
there is a scale drawn on the plan it will expand and contract with 
the paper, and if the dimensions used in calculating areas from the 
plan are measured by this scale the areas so obtained will require 
no correction. It is, however, usually more convenient to use a 
separate boxwood or other scale in taking out areas, and, if the plan 



328 



SURVEYING 



has altered, the amount of the expansion or shrinkage must be 
ascertained and an appropriate correction applied to the calculated 
area. 

Let ABCD (Fig. 265) be the size of a square with sides of length s 
as originally drawn on a plan and let Abed represent its size after 

shrinkage. Let the amount of the linear shrinkage be the fraction - 

of the original dimensions. The value of - will seldom exceed 5 Jo- 

The reduction in the length of each side of the square will be -, and, 



as shown in the figure, the reduction of area will consist of a strip - 

in width along each of two sides. 

The total reduction of area will, 

s 2s 2 
therefore, be 2s - or — ■. The 
x x 

small square portion at C where 

the sides overlap should be taken 

only once, but its amount is 

quite negligible. The contracted 

area of the square is, therefore, 

2s 2 / 2\ 
s 2 — — ors 2 1 — - , that is, the 
x \ x/' 

area of the square is reduced by 



Fig. 265.— Shrinkage of Plan. 



the fraction - of its original 



area, so that the fractional contraction or expansion of area on a 

plan is equal to twice the linear contraction or expansion. The area 

calculated from a shrunk plan must therefore be increased by the 

2 
fraction - to arrive at the true area. 
x 

Example. The area of a plot of ground is calculated from 
measurements made with the original boxwood scale used in plotting 
and works out to 12-386 acres. On comparing the boxwood scale 
with the scale drawn on the plan it is found that shrinkage has 
occurred to the extent of 1 ft. in 800 ft. What is the true area of 
the plot ? 

The fractional linear shrinkage is sua, so that the reduction 
of area is 500, and the calculated area must therefore be 



CALCULATION OF AREAS 329 

increased by or 0-031 acre. The true area is therefore 

J 400 

12417 acres. 

Areas taken out by planimeter would require to be corrected in 
the same way. 

If expansion or contraction of a plan takes place in one direction 
only the alteration of area will be fractionally the same as the 
alteration in linear dimensions. 



CHAPTER XXI 

CALCULATION OF EARTHWORK QUANTITIES 

This chapter deals briefly with the ordinary methods of calculat- 
ing earthwork quantities which are applicable to classes of work 
such as cuttings and embankments for roads, railways, canals, and 
reservoirs, excavations for trenches and foundation pits, and excava- 
tions and embankments in levelling off sites for buildings and works. 



Purpose of Earthwork Calculations. — Calculations of quantities of 
earthwork are usually made for one of the following purposes : — 

(a) In connection with the designing of works, for the purpose of 
arriving at suitable formation levels for the earthwork, that is, 
suitable levels to which the surfaces of excavations and embank- 
ments are to be finished. 

(b) For the purpose of obtaining quantities of excavation, embank- 
ment, &c, for insertion in a schedule or bill of quantities in connec- 
tion with work which is to be let out to contract. Such quantities 
are generally calculated from the information given on plans and 
sections of the proposed works. 

(c) In order to obtain final quantities for purposes of payment. 
Such quantities are measured from the dimensions of the work as 
executed or authorised. 

We have an illustration of the various purposes in the earthwork 
for a railway. A desideratum in arranging the formation line of the 
railway is that the resulting quantities of excavation and embank- 
ment should balance closely without involving long haulage. The 
usual procedure is to lay down on the longitudinal section a trial 
formation fine arranged to give equal amounts of excavation and 
embankment as nearly as can be judged by the eye, and then make a 
preliminary calculation of the quantities, the result being used to indi- 
cate the direction in which adjustment of the formation line should be 
made in order to obtain closer balance of the earthworks, if required. 

When the lines and levels of the railway have been designed, the 
quantities of excavation and embankment are calculated from the 



CALCULATION OF EARTHWORK QUANTITIES 331 



plans, sections and cross-sections. If rock is known or believed to 
occur in the cuttings, estimation is also made of the quantity of rock, 
and the quantities of soft excavation, rock excavation, and embank- 
ment are inserted in the schedule. 

As the work proceeds, measurement of the earthwork quantities 
will be made periodically in order to determine the payments due 
to the contractor, and the actual total quantities of rock excavation 
and soft excavation will be determined from measurements made 
during the progress and after completion of the work. 

The unit quantity for the measurement of earthworks in this 
country is the cubic yard (equal to 27 cub. ft.) ; where the metrical 
system of measurement is in use the unit is the cubic metre (1 cub. 
metre = 35-317 cub. ft.). 

Volumes of Solid Bodies. — The following table gives formulae 
for finding the volumes of solid bodies of various forms. The 
fundamental solids, on which the measurement of nearly all earth- 
work is based, are the prism, wedge, pyramid, and prismoid. 

Volumes op Solids. 



Figure. 



Volume = V. 




A = Area of 
N = Area of Section nor- 
mal to axis. 



Truncated Prism. 




N = Area of normal sec- 
tion. 

Ji _h 1 + h L ±hs 
ti- -3 . 

For n-sided prism h = 
h 1 +h 2 + h 3 + - • -+h n . 



332 



SURVEYING 



Figure. 



Wedge with parallel sides. 




Truncated Wedge. 




Volume = V 



V = A 



A == Area of 
& = Perp. height from 
base to edge. 



ah 
IT 



V = ^(26 + &i). 



Pyramid or Cone. 





v=a|. 

For cone with base of 

diam. d — 
v _ nd 2 h 
12 -1 



Truncated Pyramid or Cone. 




V=3(A 1 + A 2 + VA 1 A 2 ). 

V = ^(A 1 + 4A m +A 2 ) 

Prismoidal Formula. 
A„, = Area of section at 
mid -height. 



CALCULATION OF EARTHWORK QUANTITIES 333 



Figure. 



Volume = V. 



Prismoid. 




h ■■■■ 



V = g(A 1 + 4A TO + A a ). 

A x and A 2 are the end 
areas ; A m the area of 
section midway be- 
tween the ends. 



Sphere. 




Y = ^ = \nd\ 



Frustum of Sphere. 




(Sr^ + Sr^ + W). 



V = J(A 1 + 4A JIt +A 1 ). 



Prismoidal Formula. — The following is one definition of a pris- 
moid : "A prismoid is a solid having for its two ends any dis- 
similar parallel plane figures of the same number of sides, and all 
the sides of the solid plane figures also." According to this 
definition, the figure shown above under the heading " Pris- 
moid " is not a prismoid. It is shown, however, as the general type 
of solid which in the calculation of earthwork is usually known as a 
prismoid, and for which the prismoidal formula gives the exact 
volume. The prismoidal formula also gives the correct volume for 



334 



SURVEYING 



any solid having parallel plane ends and with sides formed 
by moving a straight line around the perimeters of the ends as 
directrices. It also gives the exact volume for the prism, wedge, 
pyramid, cone, sphere, spheroid, paraboloid, or for a frustum of 
any of these. 

Where the ground is of fairly uniform surface, the portion of an 
excavation or embankment, such as for a road or railway, contained 
between two parallel cross-sections a short distance apart, is for all 

practical purposes a prismoid, and its 
• K - 1° •* volume will be most accurately obtained by 

the prismoidal formula. 



Excavation in Foundation Pit. — As an 

instructive example in the fundamentals 
of earthwork measurement we may take 
the case of the excavation for a foundation 
pit or sewer manway excavated with 
sloping sides. Bottom area 5 ft. by 5 ft., 
depth 10 ft., side slopes I to 1 (4 vertical 
to 1 horizontal). 

The excavation is in the form of a trun- 
cated pyramid, but taken in detail it may 
be considered, as shown in Fig. 266, as 
made up of a central prism A, 10 ft. deep 
by 5 ft. square, four wedges B, with bases 



B 

~/6 


B 


c/ 

B 


A 


B 



Fig. 266, 



5 ft. by 21 ft. and height 10 ft., and four 
corner pyramids C, with bases 2| ft. square and height 10 ft. 

For purposes of comparison we shall work out the volume in 
detail from the several portions enumerated above, and also by 
the formula for truncated pyramid, by the prismoidal formula, 
and by approximate methods which are in common use, known 
as the " average end area " and " mean area " methods. 
In detail : 

Central prism - 10 



4x5 



X 5 X 

X2|X 



Four side wedges 

Four corner pyramids 4 X 2| X 2£ X 



5 = 250 
10 

2 

10 



cub. ft. 



= 250 



= 83J 



Total = 583^ cub. ft. 



~~ 3 


(25 + 100 + 


10 
3 


(125 + 50). 


10 
3 


X 175 = 583 



CALCULATION OF EAKTHWORK QUANTITIES 335 

By formula for truncated pyramid : 

A x = 5 X 5 = 25 sq. ft. A 2 = 10 X 10 = 100 sq. ft. 
V = ~ (25 + 100 + s/25 X 100). 



_ cub. ft. 
By prismoidal formula : 

V = ^(Ai + 4A m + A 2 ). 
A m = 7i x 7i — 56i sq . ft. 

V = ^ (25 + 225 + 100). 
__ 3jft0 __ 5g3 ^ cub _ it 

The three foregoing methods give the same result, which is the 
mathematically exact volume of the geometrical solid. 

■ By " Average End Area " Method. — By this method when Ai 
and A 2 are the two end areas and h is the distance between them, 
the approximate quantity is given by the formula V = £(A X + A 2 )h. 
In the case under consideration 

A x = 5 X 5 = 25 : A, = 10 X 10 = 100 : h = 10 

V = |(25 + 100) 10 = 625 cub. ft. 

The error in this case is + 41f cub. ft. 

By " Mean Area " Method. — The volume is in this case taken as 
being equal to that of a prism of height h and area A m equal to the 
area of a section midway between the ends, that is 

V = kji. 

k m =1l X 7| = 56£ sq. ft. : h= 10 ft. 

V = 56| X 10 = 562| cub. ft. 

The error by this method is — 20-83 cub. ft. 
A comparison of the results of the foregoing calculations shows 
that the " average end area " method gives too great a quantity, 



336 SURVEYING 

while that obtained by the " mean area " method is too small. 
The amount of the error by the " average end area " method is 
twice that by the " mean area " method, the errors in this par- 
ticular case being about + 7 per cent, and — 3| per cent, respec- 
tively. The large error in this case is due to the great relative 
difference between the end areas, the one area being four times as 
large as the other. Where the end areas are more nearly equal the 
error by the approximate methods becomes much less. Where 
one end area is 50 per cent, larger than the other the error by the 
" average end area " method is less than 1 per cent. 

Excavation and Embankment for Pond.— As a simple example 
involving the balancing of earthwork let us take the following : 
A rectangular pond is to be formed by excavating to a depth of 
7| ft. on level groi ad, the size at the bottom being 100 ft. by 120 ft., 



Fig. 267.— Earthwork for Pond. 

and the slopes of the sides 2 to 1. What is the quantity of excava- 
tion ? The excavated material is to be used to form a bank around 
the pond of a width of 40 ft. at the base and with side slopes of 
2 to 1, a bench 5 ft. wide being left all round the excavation, as 
shown in Fig. 267. To what height can the bank be formed ? 

The prismoidal formula should be used for finding the quantity of 
excavation. 

Bottom area A x = 100 X 120 = 12,000 sq. ft. 
Top area A 2 = 130 x 150 = 19,500 sq. ft. 
Middle area A^ = 115 x 135 = 15,525 sq. ft. 
71 
Volume V = -£ (12,000 + 62,100 + 19,500) 

= 117,000 cub. ft. = 4,333 cub. yards. 
Width between centre lines of banks, 130 + 10 + 40 = 180 ft. 
Length between centre lines of banks, 150 + 10 + 40 = 200 „ 
Total length of bank measured on centre line = 2 (180 + 200) 
= 760 ft. 



CALCULATION OF EARTHWORK QUANTITIES 337 



Total volume of bank = 117,000 cub. ft 

117,000 



Cross-section area of bank 



760 



= 154 sq. ft. 



Let x = height of bank. 
Then 40 — 2x — average width 
and area = x (40 — 2x) 
.-. x (40 — 2x) = 154, 
and solving the quadratic equation we get x = 5-2 ft. 



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3-6 




3-3 




8-8 




81 


7-2 




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8-1 




8-0 




7-6 




72 . 


64 


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6-0 


TO 




72 




7-2 




ff-« 


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62 


62 




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Fig. 



4-7 4-4 3-8 

-Earthwork in Levelling an Area. 



Quantity of Earthwork in Levelling an Area. — Fig. 268 illustrates 
a method of calculating the quantity of excavation or embank- 
ment involved in levelling off a building site. The calculation 
is based on the formula given on p. 331 for a truncated prism. 
The area to be levelled off is divided up into equal squares or 
rectangles of a convenient size, and such that the surface of 
the ground in each of these areas is practically a plane surface. 
Levels are taken at the corners of the areas and from these, knowing 
the final level to which the earthwork is to be formed, the depths of 
excavation or embankment at the corners of the areas are calculated 



338 SURVEYING 

and written down on a plan or diagram. If we take the area 
A.BEF on the diagram with corner heights h v h 2 , h 3 , and h i of 7-0, 
7-2, 6-2 and 6-2 ft. respectively, the volume will be given by the 
formula 

V = j (K + h 2 + h 3 + h,), 
or V = 5 ° X 5 ° (7-0 + 7-2 + 6-2 + 6-2) = 16,625 cub. ft. 



We might calculate the volume of each prism separately in this 
manner and add the results together to obtain the total volume, 
but this would be a laborious process. A little consideration will 
show that in summing the volumes of all the prisms, the depth 
at A (6-2 ft.) will have been taken only once ; the depth at B (6-2 ft.) 
which is common to two prisms will have been taken twice ; the 
depth at C, common to three prisms, will have been taken three 
times and the depth at points such as D,which is the common corner 
of four prisms, will have been taken four times, the number of times 
for each corner of the diagram being as indicated by the figures in 
small circles. We may, therefore, avoid the separate calculation 
of the volumes of the prisms if we take the sum of the corner depths 
multiplied each by its circled number, the total sum so obtained 
being multiplied by one-fourth of the base area of a prism to give 
the total volume. 

Thus for the case shown on the diagram the total volume will be 

50 X 50 
V = (9-6 + 2 x 9-3 + 2 X 8-8 + 2 X 8-1 + 7-2 + 2 

X 8-1 + 4 X 8-0 + 4 x 7-8 + 4 X 7-2 + 2 x 6-4 + 2 X 7-0 +, &c.), 
giving, when extended to include all the corners, a total of 242,687 
cub. ft. or 8,988 cub. yards. 

It will seldom happen that an area to be excavated or levelled off 
can be divided into an exact number of equal small areas as in the 
foregoing example. Where the boundaries are irregular the area 
may be divided up by two sets of parallel lines at right angles to 
each other, forming a number of rectangles in the interior of the area 
and a number of irregular figures around the boundaries. The 
volume included in all the complete rectangular prisms will be 
obtained by the method just given, and for the other irregular 
prisms around the boundary, the volume of each will be separately 



CALCULATION OF EAKTHWORK QUANTITIES 339 

obtained by multiplying its base area by the mean of its corner 
heights. 

Where the ground surface is irregular or where the earthwork is 
to be finished off with sloping sides or to an irregular formation, the 
quantity of earthworks should preferably be taken out from cross- 
sections plotted on paper to a sufficiently large scale. 



Earthwork for Roads, Railways, &c. — The general forms of the 
cuttings and embankments for railways and roads are as shown in 
Fig. 269. The formation width W and the angle of slope of the sides 
are usually constant, while 
the depth D and the inclina- 
tions of the ground surface 
vary. For a cutting or 
embankment of constant 
formation width and con- 
stant side slope, situated on 
ground which is level trans- 
versely, the area of cross- 
section can be calculated at 
any point provided the 
depth is known. Let the 
tangent of the angle which 
the slope makes with the vertical be S (for slope of 1| to 1, S = 
1J ; for slope of 2 to 1, S = 2), then 

Width at formation = W 
Width at ground = W + 2SD 
Average width = W + SD 

Area of section = D(W -f SD). 

Take an embankment for a double-line railway 30 ft. wide at 
formation, 10 ft. deep, and with slopes of 1| to 1 




Fig. 269.— Railway Cutting and 
Embankment. 



Area of section, 



X 10) = 450 sq. ft. 



We get from this a method of finding the quantities in cuttings 
and embankments from the longitudinal section, which will give 
fairly accurate results if the ground is nearly level across the line. 
The method is useful for making preliminary estimates of quantities, 

z2 



340 



SURVEYING 



and is illustrated in detail in the following example, which refers to 
the embankment shown on the longitudinal section in Fig. 270. 




"7? 73 IS I? 73 79 TO IT 

Fig. 270. — Earthwork in Kailway Embankment. 

The formation width of the double-line railway is 30 ft. and the 
side slopes are \\ to 1. The calculations are shown in the following 
table : — 

Calculation of Quantity of Embankment. 



Cross-Section. 


D 


W + 1|D. 


Area 
D(W + liD). 


Distance. 


Quantity. 




Ft. 


Ft. 


Sq. ft. 


Ft. 


Cub. ft. 


13 


0-0 


30 





50 




14 


2-8 


34-2 


96 


100 1 




15 


4-6 


36-9 


170 


100 1 




16 
17 


6-1 
6-6 


39-1 
39-9 


238 
263 


100 1 
100 f 


109,600 


18 


5-5 


38-2 


210 


100 | 




19 


3-4 


35-1 


119 


100 J 




20 


1-2 


31-8 


38 


75 


2,850 


20 50 


0-0 


30-0 





25 






112,450 






1 


^otal Quant 


ty 


f 4,165 
\ cub. yards 



In the above table the quantity is calculated on the assumption 
hat each cross-section applies to a length on either side of it extend- 



CALCULATION OF EARTHWORK QUANTITIES 341 

ing halfway to the next cross-section. The result arithmetically is 
exactly the same as if the quantities were taken out by average end 
areas, but the setting down of the figures is simplified. Where 
a series of cross-sections have to be multiplied by the same com- 
mon distance, then instead of finding the separate products and 
adding them together, add up the cross-section areas and multiply 
their sum by the common distance as is done in the foregoing 
table. 

Where the surface of the ground is sloping or irregular the quanti- 
ties will be best obtained from plotted cross-sections whose areas 
may be found by planimeter or by scaling and computation. In 
practice the average end area method of arriving at the volume is 
generally considered quite satisfactory and the prismoidal method 
is rarely applied, on account of the extra labour involved. 

Prismoidal Method. — To illustrate the prismoidal method applied 
to a continuous excavation or embankment, we shall take the 
portion AB shown on Fig. 270, which consists of three portions 
200 ft. long, each portion having two end areas and a middle area. 
We shall assume that the areas given in the table (p. 340) are the 
actual measured areas of the cross-sections. 

The prismoidal formula for a portion such as Abed is 

V=|(A 1 +4A 2 + A 3 ), 

where A l5 A 2 , and A 3 are the front, mid and end areas respectively. 
If A 3 , A 4 , and A 5 are the areas of the next portion its volume will be 

"= - (A 3 + 4A 4 + A 5 ) and so on for successive portions. If these 

volumes are added together we get the general formula for the 
total volume 

V = - (A x + 4A 2 + 2A 3 + 4A 4 + 2A 5 , + . . . + AJ 

where h is the length of each assumed prismoid, that is twice the 
interval between cross-sections. If I is the common interval 



between sections U — - j , the formula will be 

V = 3 (A 1 + 4A 2 + 2A 3 + 4A 4 + 2A 5 + . . . + AJ. 



342 SURVEYING 

Applying the formula to the portion AB we have 

100 
V = — (96 + 4 X 170 + 2 X 238 + 4 X 263 + 2 X 210 + 4 X 

100 
119 + 38) = — x 3,238 = 107,933 cub. ft. 

96 X 100 
End portion at A = ~ = 4,800 cub. ft 



50 
End portion at B = 38 X ^ = 950 



Total embankment = 113,683 cub. ft. 

= 4,210 cub. yards. 



2 ~ -,— —.«. 



CHAPTER XXII 

ADJUSTMENT OF INSTRUMENTS 

Dumpy Level. — In all forms of Dumpy level there is provision for 
adjusting the line of sight by raising or lowering the diaphragm 
containing the cross hairs. Generally, also, there is provision for 
making adjustment between the bubble tube and the telescope and 
for making adjustment between the telescope and the stage, as 
indicated in Fig. 195. In some levels, however, the bubble tube is 
fixed with respect to the telescope, and in others the telescope is 
fixed with respect to the stage, or, what amounts to the same thing, 
the stage is dispensed with and the telescope is directly and solidly 
connected to the vertical axis. 

The purpose of making adjustments is that the following require- 
ments of a correct level may be attained : — 

(a) The line of sight of the telescope should be parallel to 
the bubble axis. 

(b) Both the line of sight of the telescope and the axis of the 
bubble should be at right angles to the vertical axis of rotation 
of the level. 

In the case of the ordinary Dumpy level, with means of adjust- 
ment provided at three points, as above mentioned, the attainment 
of requirement (a) can for most practical purposes be effected by 
making adjustments at any two of the three points. It is better, 
however, to avoid altering the cross hairs, assuming that these have 
been correctly placed by the maker so that the line of sight and line 
of travel of the centre of the object glass are in coincidence. The 
adjustment of the ordinary Dumpy level to effect compliance with 
requirement (a) are, therefore, made by manipulation of the 
screws connecting the bubble tube to the telescope and the screws 
connecting the telescope to the stage. 

Peg Method of Adjustment to make the Line of Sight parallel to the 
Bubble Axis. — The process known as the peg method is illustrated 
in Fig. 271. The level is planted midway between two pegs or 



344 



SURVEYING 



firm, definite points A and B, say, 400 ft. apart, and levelled up. 
Readings are taken to a staff held at points A and B, the bubble 
being brought exactly to the centre of its run for each reading. 
The difference of these readings will give the true difference of 
elevation of the two points, as although the line of sight may not be 
horizontal, the error in the readings will be the same in each case 
since the distances are equal. 

Referring to Fig. 271, the lines of sight from the level at position 
Cj will cut the staffs at points a and b, which will be at the same level, 
so that the difference of the readings A« and B6 will be the correct 
difference of level of the points A and B. The instrument is now 
transferred and set up at a point C 2 , a short distance beyond one of 
the two pegs and nearly in line with them. Make the distance 




£00' >|< 2Q0' ... 

Fig. 271. — Adjustment of Level. 



x-50' -> 



BC 2 an even fraction of the distance AB, say, 50 ft. The instru- 
ment having been levelled and the bubble brought to the centre 
readings are again taken to the staffs at A and B, the readings 
being represented by Ao x and B6 : respectively. If the level is in 
correct adjustment, points a x and b x will be at the same level, or 
D x and D 2 , the differences of the staff readings on A and B respec- 
tively, will be the same. If D x and D 2 are not the same the instru- 
ment is incorrect and the amount of error of the line of sight in the 
length AB will be equal to D 1 — D 2 . By proportion we get the 

AC 
amount of error in the distance AC 2 as equal to (D x — D 2 ) X -^5 . 

AB 

Similarly, the amount of error in the length BC 2 is equal to (D x — D 2 ) 

BC 2 
X t^5". The amounts are represented by the distances aji and b x k 



ADJUSTMENT OF INSTKUMENTS 345 

on the figure respectively. If we subtract the error aji, calculated 
as above from the staff reading Aa 1} we get the reading Ah, which 
a correct level would give. 

The levelling screws are, therefore, turned so as to make the 
cross hair read the calculated value Ah. This will have the effect 
of throwing the bubble out of centre. The bubble is brought back 
to the centre of its run by raising or lowering one end by means of 
the adjusting screws connecting its casing to the telescope, care 
being taken not to disturb the reading on the staff. A reading 
should also be taken on the staff at B, and if the value obtained is 
equal to the former reading B6 X increased or diminished by the 
calculated error b-Jc, the bubble remaining central, we may take it 
that the requirement (a) has been fulfilled. 

An example worked out will make the method more explicit. 

Points A and B were 400 ft. apart, and C 2 was 50 ft. beyond B. 
The readings and calculations were as follows : — 

Reading on A. Reading on B. 

Level at C x . 5-28 3-75 

Level at C 2 . 7-51 5-66 

B 1 = — 2-23 D 2 = — 1-91 

J) 1 — D 2 = 0-32 = Error in AB. 

450 
Error in AC 2 = 0-32 X ttt = 0-36 

Correct staff reading at A = 7-51 — 0-36 = 7-15 

. ' 0-32 X 50 
Error m BC 2 = — — = 0-04 

Correct staff reading at B = 5-66 — 0-04 = 5-62. 

The telescope is, therefore, tilted by means of the levelling screws 
till the reading on staff at A is reduced from 7-51 to 7-15. The 
line of sight is thus made horizontal and the bubble is then brought 
to the centre of its run by means of its adjusting screws. 

If, in making the subtractions to obtain the differences of the 
staff readings, Dj^ and D 2 come out of opposite sign, the error in 
length AB will be the arithmetical sum of the values D x and D 2 . 

To make the Line of Sight and Bubble Axis perpendicular to the 
Vertical Axis. — The further adjustment required to make the 



346 SURVEYING 

line of sight and the axis of the bubble perpendicular to the 
vertical axis is effected by means of the screws connecting 
the telescope to the stage. The instrument is levelled up, the tele- 
scope is placed over a pair of opposite screws, and by turning these 
the bubble is brought exactly to the centre. The telescope is 
rotated through 180° on the vertical axis so as to point in the opposite 
direction. If the bubble is still central its axis is at right angles to 
the vertical axis. If the bubble has deviated from the centre it 
should be brought halfway back by the levelling screws and the 
rest of the way by the adjusting screws connecting the telescope to 
the stage. The process will be repeated and further slight adjust- 
ment made, if necessary, till the bubble is found to remain central 
when the instrument is turned in any direction. 

Adjustments when the Bubble is permanently fixed to the Telescope. 

— In this case it is better to do first the adjustment for making the 
bubble axis perpendicular to the vertical axis by the method de- 
scribed in the preceding paragraph. The peg method is used for 
making the line of sight parallel to the bubble axis. Referring to 
Fig. 271, after the readings on the pegs have been taken from 
position C ls the instrument is set up at C 2 and levelled accurately. 
Readings are taken to the staffs at A and B and if this shows that 
the line of sight is not horizontal, the correction is made by raising 
or lowering the diaphragm containing the cross hairs. If, as in the 
example worked out, a lower reading is required on the staff at A 
to give a horizontal line of sight then the diaphragm must be raised ; 
to give a higher reading the diaphragm must be lowered. To raise 
the diaphragm the lower capstan screw must first be loosened and 
the upper screw then tightened, and vice versa to lower the dia- 
phragm. When the line of sight has been altered so as to give the 
true difference of elevation of the pegs the adjustment is complete. 
Care must be taken to see that the bubble is kept central. 

Adjustments when the Telescope is firmly fixed to the Vertical Axis. 

■ — In this case the bubble axis must first be made perpendicular to the 
vertical axis. This is done by setting up and levelling the instru- 
ment, placing the telescope over a pair of screws, bringing the bubble 
exactly to the centre, reversing the telescope end for end and cor- 
recting the displacement of the bubble one half by the levelling 
screws and the other half by the adjusting screws connecting the 



ADJUSTMENT OF INSTRUMENTS 347 

bubble to the telescope, the process being repeated till the bubble 
remains central for any position of the telescope. 

The remainder of the adjustment, to make the line of sight 
parallel to the bubble axis is accomplished by the peg method and 
the raising or lowering of the diaphragm exactly as described in the 
foregoing paragraph. 

Wye Level. — In the Wye level the diaphragm containing the 
cross hairs is adjustable in two directions at right angles to each 
other, the means of adjustment being provided in order that the 
line of sight of the telescope may be made to coincide with the 
axis of the circular collars of the telescope. 

The adjusting screws connecting the bubble tube to the tele- 
scope enable the axis of the bubble tube to be made parallel to 
the supporting surfaces of the wyes on which the collars rest. If 
the collars are of equal diameter, as they should be, the bubble 
axis will then be also parallel to the axis of the collars. 

The means of adjustment provided between the telescope and 
the stage enables the bubble axis and line of sight to be made 
perpendicular to the vertical axis of the instrument. 

Adjustment of the Cross Hairs. — Also known as the adjustment 
for collimation. The telescope is sighted and focussed on to a well- 
defined point, and carefully adjusted by means of the levelling 
screws till the intersection of the cross hair with the vertical hair 
is exactly on the point. The stage is clamped, the clips are 
loosened, and the telescope is rotated halfway round so as to bring 
the bubble from the top position to the bottom or vice versa. If the 
intersection of the hairs has deviated from the mark, bring the 
horizontal cross hair halfway back by means of the vertical pair of 
diaphragm capstan screws and the vertical cross hair halfway back 
by means of the horizontal pair of screws. Then sight again on the 
mark and test by again turning the telescope half round, and 
adjusting, if necessary, till the centre of the cross hairs appears to 
remain fixed, as the telescope is rotated through a revolution. 
The line of sight is then parallel to and practically in coincidence 
with the axis of the circular collars of the telescope. Where 
there are two vertical cross hairs and a single horizontal hair 
giving two intersections, both intersections should describe a circle 
about the point midway between them as centre. 



348 SURVEYING 

To make the Axis of the Bubble Tube parallel to the Line of Supports 
of the Wyes. — The instrument having been levelled up, the telescope 
is placed over a pair of screws and clamped in position and the 
bubble is brought exactly to the centre. The clips are then loosened, 
the telescope is taken out and reversed end for end in the collars. 
If the bubble is now displaced from the centre the axis of the tube 
is not parallel to the supports. The correction is made by bringing 
the bubble halfway back by the levelling screws and the rest of the 
way by the capstan screws connecting the bubble tube to the 
telescope. Having made the adjustment, repeat the test and 
correct again if necessary. 

The adjustment of the bubble tube and the adjustment of the 
cross hairs having been made, the line of sight and the axis of the 
bubble tube will now be parallel, provided the collars are of the same 
diameter. If there is any doubt as to this the adjustment to make 
the bubble axis parallel to the line of sight should be made by the 
peg method, as described for the Dumpy level. 

To make the Axis of the Bubble Tube perpendicular to the Vertical 
Axis of the Instrument. — The instrument having been set up and 
levelled, the telescope is placed over a pair of levelling screws and 
the bubble is brought exactly to the centre. Rotate the telescope 
through 180° about the vertical axis so that it is turned end for end 
over the same pair of screws. If the bubble has deviated from the 
centre the axis of the bubble tube is not at right angles to the 
vertical axis. To make the correction, bring the bubble halfway 
back by the levelling screws and the remainder of the way by the 
adjusting screws connecting the wyes to the stage. Repeat the 
test and again adjust, if necessary, till the bubble is found to 
remain central for any position of the telescope. 

Theodolite. — The following adjustments are required in a theodo- 
lite to be used for reading horizontal angles : — ■ 

(1) The adjustment of the levels on the vernier plate so that 
the axes of their bubble tubes shall be perpendicular to the 
vertical axis of the instrument. 

(2) The adjustment of the cross hairs so that the line of sight 
of the telescope shall be perpendicular to the transit axis and shall 
thus describe a plane as the telescope is transited. 



ADJUSTMENT OF INSTRUMENTS 349 

(3) The adjustment of the supports on top of the standards so 
that the transit axis shall be truly horizontal and the line of sight 
shall revolve in a vertical plane when adjustments Nos. (1) and (2) 
have been effected and the instrument has been levelled up. 
The following additional adjustments are required if the instru- 
ment is to be used for levelling and for reading vertical angles : — 

(4) The adjustment of the level on the telescope so that the 
axis of its bubble tube shall be parallel to the line of sight. 

(5) The adjustment of the vernier arms of the vertical circle 
so that the reading shall be zero when the line of sight is horizontal. 

Adjustment of the Plate Levels so that the Axes of their Bubble 
Tubes shall be perpendicular to the Vertical Axis of the Instrument. — 

Set up the instrument on firm ground with the lower clamp loose 
and the upper clamp fixed. Set the longer of the two levels parallel 
to a pair of diagonally opposite screws, or in the case of a three-screw 
instrument over any pair of screws. Turn the levelling screws till 
both bubbles are brought to their centres. Then turn the head of 
the instrument through 180° so that each level is reversed end for 
end and is over the same screws as before. If the bubbles remain 
central the levels are in correct adjustment. If not, adjust first the 
longer level by raising or lowering one end by means of the capstan 
nuts (c, c, Fig. 127) which connect it to the vernier plate, so as to 
bring the bubble halfway back to the centre, the bubble being 
brought back the rest of the way by turning the levelling screws. 
Repeat the test for this level by turning the head back through 180° 
and noting if the bubble now remains central, and adjust again if 
necessary. When the longer level has been brought to satisfactory 
adjustment in this manner the shorter one may be simply adjusted 
by raising or lowering one end by means of its capstan nuts till 
the bubble is central simultaneously with the bubble of the longer 
level. 

Adjustment of the Cross Hairs. — The diaphragm of the theodolite 
telescope is usually fixed by two pairs of capstan screws, one in the 
vertical direction and the other in the horizontal direction. Some- 
times, as in the instrument shown in Fig. 127, the cross hairs are 
adjustable only in the horizontal direction. For the reading of 
horizontal angles, prolonging of straight lines and setting out of 



350 SURVEYING 

works the accurate adjustment of the cross hairs in the horizontal 
direction is of prime importance. 

To make the horizontal adjustment set up and level the instrument 
and sight the intersection of the cross hairs exactly on a distant and 
definite back mark. Transit the telescope and line out an arrow 
at a considerable distance, say, 300 ft. from the telescope. Loosen 
the lower clamp, turn the head of the instrument through 180°, 
sight on the back mark and again transit the telescope. If the line 
of sight strikes the arrow the cross hairs are in correct horizontal 
adjustment. If the line of sight deviates from the arrow line out 
a second arrow opposite the first and measure the distance between 
them. The correction of the hairs is made by putting in an inter- 
mediate third arrow at one-fourth of the deviation measured back 
from arrow No. 2 and bringing the line of sight to strike this arrow 
by adjusting the diaphragm horizontally. If the line of sight 
requires to move towards the left in shifting from arrow No. 2 to 
arrow No. 3 the diaphragm will require to be moved to the right 
and vice versd. The motion of the diaphragm is effected by first 
loosening one of the screws (b, b, Fig. 127) and then tightening the 
other. Having made the adjustment sight again on the back mark 
and transit the telescope. The line of sight should now strike 
midway between arrows Nos. 1 and 2. If it does so the horizontal 
adjustment is correct. If not, repeat the test and make further 
slight correction till the same line of sight is given on transiting with 
the telescope normal as with the telescope inverted. 

Another method of making this adjustment is as follows : The 
instrument having been accurately levelled sight on a fine mark, 
such as the point of an arrow, at a considerable distance, say, 
300 ft., making the centre of the cross hairs cut the mark exactly 
by turning the lower tangent screw. Loosen the clips which hold 
the ends of the transit axis in their bearings, loosen the opposing 
screws which fix the vertical clipping arm to the standard, carefully 
lift the telescope and replace it with the ends of the transit axis 
reversed. Transit the telescope so as to point again to the sighting 
mark. If the centre of the cross hairs hits the mark the adjustment 
is correct. If there is displacement, correct half the error in this 
case by the pair of horizontal capstan screws (b, b, Fig. 127), and 
then repeat the test and make further correction, if necessary, till 
the adjustment is satisfactory. If it is necessary to perform this 



ADJUSTMENT OF INSTRUMENTS 351 

adjustment in a limited space a fine black mark on a piece of white 
paper fixed at a distance of 30 or 40 ft. will serve quite well as a 
sighting mark. 

Adjustment of the Supports at Top of the Standards. — Set up and 

level the instrument in a position where some high definite point 
can be sighted, giving a considerable angle of elevation. With 
telescope normal sight to the point, depress the telescope and fine in 
an arrow on the ground or make a mark somewhere on the fine of 
sight nearly under the high point. Sight again to the high point 
with telescope inverted, depress the telescope and note if the line 
of sight strikes the arrow or mark. If it does so the supports are in 
correct adjustment and the line of sight revolves in a vertical plane. 
If the line of sight with telescope inverted deviates from the arrow 
or mark, put a second arrow or mark into fine opposite the first one 
and plant a third arrow midway between them. A plane through 
the high point, the axis of the instrument and this third arrow will be 
a vertical plane, and the adjustment consists in raising or lowering 
the adjustable support so as to make the fine of sight revolve in this 
vertical plane. The telescope is sighted to the third arrow and 
tilted up to view the high point. The adjustable support is raised 
or lowered by means of the opposing screws (a, a, Fig. 127) till the 
intersection of the hairs cuts the high point or is exactly on the 
vertical fine through the point. The adjustment should then be 
correct, as shown by the fine of sight striking the middle arrow when 
the telescope is again depressed. Repeat the test and make further 
slight correction if required. 

Adjustment of the Axis of the Telescope Level parallel to the Line 
of Sight. — This adjustment is effected by the peg method in the 
manner described for the Dumpy level. 

Adjustment of Vertical Circle Verniers. — In order that an angle 
of elevation or depression from a horizontal plane may be read 
directly it is necessary that the verniers should indicate zero when 
the line of sight is horizontal. Set up and level the instrument with 
the telescope over a pair of screws. Set the vernier index of the 
vertical circle to zero. If the telescope bubble is not central, turn 
the. opposing screws attaching the vertical clipping arm of the 
vernier to the standard till the bubble is brought exactly to the 



352 SURVEYING 

centre of its run. The motion of the opposing screws on the clipping 
arm causes the vernier arms, vertical circle and telescope to rotate 
in the vertical plane so that the vernier reading remains at zero, and 
when the bubble is brought to the centre the line of sight is horizontal 
and an angle of elevation or depression may be read from that 
position. The adjustment is complete when the telescope bubble 
remains central, as the head of the instrument is rotated. To effect 
this, having got the bubble central for one position, as above 
described, rotate the head through 180°. The bubble may now 
have departed from the centre. If so, correct half the error by the 
levelling screws and the other half by the opposing screws attaching 
the clipping arm to the standard. Perform the operation again, start- 
ing with the telescope at right angles to its first position and repeat, 
if necessary, till the bubble remains central as the head is rotated. 

In some instruments, such as Cooke's pattern of transit theodolite, 
illustrated in Fig. 127, the long sensitive level, instead of being 
placed on the telescope is attached to the vernier arm. In this case 
the level must be adjusted so that " when the central line of vision 
of the telescope is horizontal, and the zero lines of the vertical 
verniers coincide with the zero diameter of the vertical circle, the 
bubble may be in the middle of its run." The following is the 
method of adjustment recommended by Messrs. Cooke : — 

Having levelled the instrument carefully by means of the bubbles 
on the horizontal plate, bring the bubble in the azimuth level to the 
middle of its run by means of the antagonistic screws e, e, at the 
end of the clipping arm. Now set the zero diameter of the vertical 
circle to coincide exactly with the zero lines on the vertical verniers 
and clamp it there. Observe an ordinary levelling staff held at as 
great a distance as it can be distinctly seen, and take the reading 
by the horizontal web. Now release the clamp and transit the 
telescope and again adjust the zero diameter of the vertical circle 
to the zero lines on the verniers. Revolve the head in azimuth one- 
half turn, bringing the telescope to its former position, and once more 
take the reading of the staff. If it is not the same as previously 
observed correct half the error by the antagonistic screws at the end 
of the clipping arm and then repeat the operation until all error 
is by this means eliminated. When the adjustment is complete 
correct the azimuth level by means of the capstan headed locknuts 
d, d, so that the bubble remains in the middle of its run. 



APPENDIX. 

Geometric and Trigonometric Formulae. 

Eight Angle Triangle. 
B 




a 2 + 6 2 . 



c = v /a 2 + & 2 . 

62 = C 2 _ fl 2 # 



& => /{c-\-a){c — a). 



5. a 2 = c 2 - & 2 . 

6. a == J{c + 6) (c — 6). 

7. A + B = C = 90°. 

8. A = 90° - B. 



9. 


Sin A = - = cos B. 
c 


15. 


a ■=■ c sin A = c cos B. 


10. 


Cos A = - = sin B. 
c 


16. 


a = b tan A = b cot B. 


11. 


Tan A =% = cot. B. 




17. 


& = c sin B = c cos A. 


12. 


Cosec A = - = sec B. 
a 


18. 


b = a tan B = a cot A 


13. 


Sec A = t = cosec B. 


19. 


— cos A — sin B' 


14. 


Cot A = - . = tan B. 
a 


20. 


a a 


~ sin A — cos B" 



21. 
22. 
23. 



Oblique Triangles. 

A + B + C = 180°. B 

a b _ c 

sin A ~~ sin B — sin C 
a = b cos C + c cos B. 
b = C cos A -{- a cos C. /) 
c = a cos B + b cos A. 




354 



APPENDIX 



24. a 1 = 6 2 + c 2 - 26c cos A. 

a sin C 



25. tan A = 



b — a cos C. 



0R ■ A /( 8 -b)(8-C) 

26. sin -„ = y - 



6c 



, where 5 = ^ (a + b -f c). 



27. cos^ = x / 



s (s — a) 



be 



28. tanW ( 5 -f )( *- c) . 

2 v s (s —a) 



Solution of Oblique Triangles. 



Given. 


Required. 


Formulae. 


A,B 

a, 6, c 


c 

A 

c 
A 

B 

Area 
B 

C 
c 


C = 180° - (A + B). 

Nos. 26, 27, or 28. 

The calculation is most easily made by No. 


C, a, b 
A, a, b 


_ A /s(s — a) 
27, cos 2^ = V 6c ' 

Sin A=A */s{s — a) (s — 6) (s — c). 
be v 

No. 24, c == Va 2 + 6 2 — 2a6 cos C. 

_. , a sin C 
Sm A = ____-. 

Sin B = &E5.9, or B = 180° - (A + C). 
c 

A = J ab sin C. 

. ^ 6 sin A 

Sm B = . 

a 

Sin C = C Shl A orC- 180° - (A + B). 
a 

a sin C 

c = — — J". 

sm A 
Note that B may have two values, one 
less than 90° and the other greater than 90°, 
hence C and c may each also have two 
values. 



APPENDIX 



355 



Given. 


Required. 


Formulae. 


B, C, a 


A 

b 

c 
Area 


A - 180° - (B + C). 

, a sin B 
b = — — t-. 

sin A 

a sin C 
c = — — t-. 
sin A 

a 2 sin B sin C 

2 sin A 



A A 2 



INDEX 



A. 



Abney level, 23. 267, 290 

Adjustment of : 

angles in triangulation, 201 

angular errors in traverse, 175 

closing error in traverse, 164, 176 

dumpy level, 343 

latitudes and departures, 176 

theodolite, 349 

Wye level, 347 

Altitudes by theodolite, 219, 221 

Amsler planimeter, 323 

Angles : 

adjustment of, in triangulation, 201 
booking, for traverse survey, 151 
direct angle method of measuring, 

145 
errors in reading, 203, 206 
laying off by chord method, 156 
laying off by protractor, 154 
laying off by tangent method, 155 
measuring by doubling, 140 
measuring by repetition, 140, 199 
measuring by series method, 200 
measuring horizontal, 137, 198 
measuring vertical, 139 
methods of reading, in traverse 

survey, 145 
of polygon, check on, 147 
separate angle method of measur- 
ing, 145 
setting out with chain or tape, 72 
whole circle bearing method of 
measuring, 148 

Angular errors in traverse, adjustment 
of, 175 

Areas : 

British units of, 313 

metrical units of, 313 

of geometrical figures, 314 

Areas, calculation of : 

by equalising boundaries, 318 

by equivalent triangles, 317 

by mean ordinate rule, 321 

by offsets, 320 

by parallel strips, 318 

by planimeter, 322 



Areas, calculation of- -continued. 

by Simpson's rule, 322 

by trapezoidal rule, 321 

from field measurements of tra- 
verse, 326 

from survey plan, 316 
Arrangement of : 

plan, 88 

survey lines, 55, 112, 143 
Arrows, 14, 34 
Attraction, local, 103 
Axis of bubble tube, 227 



B. 



Back bearing, 105 

Band, steel, 12 

Band, testing, 31 

Base line, in triangulation : 

broken, 192 

enlarging, 193 

measuring, 190 

plotting, 89 
Bearing of line : 

back, 105 

calculation of, from direct angles, 
146 

finding from co-ordinates, 213 

finding from latitudes and de- 
partures, 212 

forward, 105 

laying off by protractor, 154 

laying off by tangent method, 155 

magnetic, 105 

true, 105 
Bench mark, 239 
Boat, distance to shore from, 211 
Bolts, 15 
Booking : 

angles of traverse survey, 151 

cross sections, 270 

survey lines, 50, 114 
Boundaries : 

conventions for, 94 

locating irregular, 47 
Box sextant, 114 
Bridge, setting out, 311 



358 



INDEX 



British units : 

linear measure, 10 

square measure, 313 
Broad needle, 102 
Bubble tube : 

adjustment of, 345 

axis of, 227 
Buildings : 

conventions for, 97 

fixing position of irregular, 46 

fixing position of rectangular, 45 

setting out, 309 

C. 

Calculation of 

areas, 313 

bearing from direct angles, 146 

earthwork quantities, 330 

levels, 240 
Chain, steel, 11, 31, 33 
Chain surveying : 

arrangement of survey lines, 55 

errors in, 61 

expedition in, 67 

field operations, 26 

instruments, 10 

in towns, 80 

plotting, 81 

running a survey line, 40 

special problems, 69 
Chaining : 

correction for length on slope, 37 

correction on ground, 38 

on level ground, 33 

on sloping ground, 36 

overcoming obstacles to, 72 

past inaccessible area, 73 

past obstructions, 75 

precautions in, 35 

stepping, 36 

survey lines, 31 
Check on angles of polygon, 147 
Chord method of laying off angles : 

on ground, 72 

on paper, 156 
Closing error in traverse : 

graphical adjustment of, 164 

limits of, 184 

significance of, 184 
Coloured lines, 94 
Colouring, 99 
Compass : 

beam, 85 

drawing, 84 

graduations of, 108 

on level, 232 

on theodolite, 109, 125 

paper strip as substitute for, 85 

prismatic, 108 



Compass — -continued. 

surveyor's, 106 

taking magnetic bearing with, 110 

trough, 125 
Compass surveying : 

booking, 114 

use of, 111 

with free needle, 112 
Contour grading, 289 
Contours : 

indicating depression, 275 

locating, 276, 278 

plotting sections from, 274 

typical plan of, 273 

use of, 271 
Contour maps, use of, 275 
Contour plans, use of : 

in calculation of quantities, 283 

in laying out building areas, 282 

in locating route, 275, 280 
Conventional signs, 93 
Convention for ; 

boundaries, 94 

buildings, 97 

fences, 94 

land, 97 

railways, 94 

roads, 94 
Cooke theodolite, adjustment of, 352 
Co-ordinates of traverse, calculation of, 

172 
Correction for local attraction, 113 
Cross hairs, 128, 230 
Cross hairs, adjustment of, 347, 349 
Cross-hatching, 100 
Cross sections : 

booking, 270 

purpose of, 260 

with inclinometer, 267 

with level, 267 

with mechanic's level, 269 
Cross-section paper, 271 
Cross staff, 15 
Curvature of earth : 

as to neglecting, 7 

in finding altitudes, 8, 220, 255 

in surveying, 6 
Curves, manufactured, 85 
Curves, setting out 

by offsets from chords produced, 
297 

by offsets from tangent, 293 

by offsets scaled from plan, 292 

by radius, 293 

by theodolite, 301 

changing position of theodolite, 303 

overcoming obstructions, 309 

problems, 299 

to left, 307 
Curve tables, Krohnke's, 306 



INDEX 



359 



Datum, 239 

Declination, magnetic, 103 

Deflection angle, 302, 306 

Departures : 

calculation of, 169 

finding from traverse tables, 171 

in surveying, 168 

Details, plotting, 91 

Diaphragm, 128, 230 

Difficulties of surveying, 9 

Distance : 

across a river, 77 

from boat to shore, 211 

to inaccessible points, 210, 211 

Dividers, 85, 87 

Dotted lines, 94 

Doubling angles, 140 

DraAving paper, 81 

Drawing pen, 87 

Dumpy level, 228, 230 

Dumpy level, adjustment of, 343 



Earthwork quantities : 

by average end areas, 335 

by mean areas, 335 

by prismoidal method, 341 

in foundation pit, 334 

in levelling an area, 337 

in pond, 336 

in roads and railways, 339 

purpose of, 330 

Edge bar needle, 102 

Erasing lines, 93 

Errors in chaining : 

chain not horizontal, 63 

chain not straight, 63 

from fixing arrows, 64 

from length of chain, 61 

from marking chain lengths, 64 

from ranging lines, 65 

from sag of chain, 64 

in booking measurements, 66 

in locating objects, 65 

in reading chain, 35, 64 

permissible, 66 

Errors in levelling : 

booking the readings, 254 
carelessness in use of level, 252 
curvature, refraction, etc., 255 
faulty adjustment, 251 
mistakes in reducing, 257 
permissible, 258 
reading the staff, 254 
staff and its use, 252 

Errors in plotting, 91 



Errors in reading angles : 

displacement of signal, 205 
faulty adjustment, 206 
incorrect bisection of signal, 205 
incorrect focussing, 204 
incorrect levelling, 204 
natural causes, 205 
planting the theodolite, 204 

Errors in traverse, 164, 177 
closing, 184 

Example of survey plan, 97, 263 

Excavation, quantity of : 
in foundation pit, 334 
in levelling an area, 337 
in pond, etc., 336 

Expedition in surveying, 67 

Eye, human, 24 

Eye-piece, 127 



Fences, conventions for, 94 

Field book, 49, 245 

Field standard for testing chain, 32 

Field survey notes, 50 

Fixing buildings from survey line, 44 

Flags. 15 

Focussing : 

error from incorrect, 204 

eye-piece, 127 

object glass, 127 
Follower in chaining, 33 
Formulae, geometric and trigonometric, 

353 
Forward bearing, 105 
Four screw levelling arrangement, 123, 

230 
Full lines, 93 

G. 

Geometric formulae, 353 
Graduated circle, 124 
Graduations of compass, 109 

H. 

Hatching, cross, 100 
Heights by theodolite, 219 
Horizontal circle, 124 
Horizontal projections of lengths, 9 



Inaccessible area, chaining past, 73 
Inaccessible points : 

distance between, 211 

distance to, 210 



360 



INDEX 



Inclinometer, 22, 267 

Inking-in, 92 

Instrument height method of reducing 

levels, 241, 247 
Intersection point : 

in setting out curves, 304 

inaccessible, 307 



Land : 

area of, 316 

conventions for, 97 
Laths, 15, 277, 290 
Latitudes : 

calculation of, 169 

finding from traverse tables, 171 

in surveying, 167 
Latitudes and departures, adjustment 

of, 176 
Leader in chaining, 33 
Length : 

of offsets, 48 

of sight, 257 
Lettering, 97 

Level and staff, use of, 236 
Levelling : 

across pond, etc., 287 

appropriate length of sight, 257 

booking readings, 245 

checking levels, 247, 249 

continuous, 242 

on steep slope, 284 

overcoming obstructions, 286 

overhead point, 285 

over summits and hollows, 284 

permissible error in, 258 

purpose of, 224 

reciprocal method, 288 

reducing levels, 245 

setting up instrument, 237 

signals, 238 

trigonometric, 220 
Levelling instruments : 

Abney level, 23 

dumpy level, 228, 230 

mechanic's level, 227 

spirit level, 226 

staff, 233 

Troughton and Simms' level, 228, 
233 

water level, 225 

Wye level, 228, 233 
Levelling-off an area, 337 
Levelling-up theodolite, 137 
levels, calculation of : 

instrument height method, 241 

rise and fall method, 240 



Level surface, 224 
Line : 

bearing of, 104 

finding length from co-ordinates, 
213 

finding length from latitude and 
departure, 212 

prolonging with theodolite, 141 

ranging by eye, 26 

ranging by theodolite, 140, 142 
Linear measurement : 

British units, 10 

metrical units, 1 1 

rules for conversion, 11 
Line of collimation, 128 
Line of sight, 128 
Line of sight, adjustment of, 343 
Line ranger, 21 
Lines on paper, 93, 94 
Local attraction, 103, 113 
Locating boundaries, 43, 47 
Longitudinal section : 

of railway, 263 

of sewer, 262 

plotting, 263 

purpose of, 260 



M. 



Magnetic bearing : 

of a line, 105 

taking, with compass, 110 
Magnetic : 

declination, 103 

meridian, 102 

needle, 101 

north, 102 
Marking survey stations, 59, 197 
Mean ordinate rule for areas, 321 
Measuring : 

angles by repetition, 140, 199 

angles by series method, 200 

base line, 190 

horizontal angles, 137, 198 

horizontal projections of length, 9 

offsets, 40 

ties, 40 

vertical angles, 139 
Measuring instruments : 

chain, 11 

linen tape, 12 

steel band, 12 

steel tape, 12 

Avooden rods, 14 
Meridian, magnetic, 102 
Metrical units : 

of area, 313 

of length, 10 
Model of surface of ground, 1 



INDEX 



361 



Nails, 15, 59 

Needle : 

broad, 102 
edge bar, 102 
magnetic, 101 

North, magnetic, 102 



0. 



Object glass, 126 
Obstructions, overcoming : 

in chaining, 72 

in levelling, 286 

in setting out curves, 309 
Official standard for testing chain, 31 
Offsets : 

in surveying, 40, 43, 44 

length of, 48 

plotting, 90 
Offset staff, 12 

Omitted side of traverse polygon, 213 
Optical prism, 20 
Optical square, 16 
Ordnance datum, 239 
Ordnance scales, 82 



Papek, drawing, 81 

Paper strip, substitute for compasses, 

85 
Parallel curve, setting out, 309 
Parallel line : 

setting out with chain, etc., 71 

setting out with theodolite, 209 
Parallel plates, 123, 231 
Parallel ruler, 87 
Pegs, 15 

Peg method of adjustment, 343 
Pen, drawing, 87 
Pencilling, 92 
Pencils, 81 

Permissible error in chaining, 66 
Perpendiculars, setting out : 

on paper, 181 

with chain, etc., 69 

with optical square, 18 

with theodolite, 208 
Plan, arrangement of, 88 
Plane surveying, 1 
Planimeter, 322 
Plate levels : 

adjustment of, 349 

of theodolite, 125 



S. 



Plotting : 

base line, 89 

by co-ordinates, 166, 180 

errors in, 91 

main triangles, 89, 202 

offsets, 90 

ties, 91 

traverse survey, 153, 166, 180 

triangulation stations, 202 
Plummet or plumb-bob, 135 
Pocket dividers, 87 
Poles, ranging, 14 
Pond : 

earthwork in, 336 

levelling across, 287 

surveying, 78 
Precautions in chaining, 35 
Pricker, 82 

Principles of surveying, 2 
Printing, 97 
Prismatic compass, 108 
Prismoid, 333 

Prismoidal formula, 332, 333. 341 
Prism square, 20 
Prolonging a line : 

by eye, 28 

by theodolite, 141 
Proof line, 56 



R. 



Railway : 

convent. ons for, 94 

earthwork in, 339 

setting out. 264, 290, 304 

working section of, 263 
Ranging poles, 14 
Ranging straight lines : 

across hollow, 29 

by eye, 26 

by theodolite, 140 

over hill, 30 

past obstacles, 31, 142 
Reciprocal levelling, 288 
Repetition method of measuring angles, 

140, 199 
Right angle, setting out : 

on paper, 181 

with chain, etc., 69 

with optical square, 18 

with theodolite, 208 
Rise and fall method of reducing levels, 

241, 245 
River 

distance across, 77 

levelling across, 287 

surveying far side of, 78 



B B 



362 



INDEX 



Road : 

conventions for, 94 

earthwork in, 339 

setting out, 290 
Ruler, parallel, 87 
Rules for conversion of British to 

metrical units, 11 
Running longitudinal section, 201 
Running survey line, 40, 48 



Scales 

for plotting surveys, 82 

ordnance, 82 
Scotch staff, 234 
Sections, 260 
Separate angles : 

measuring, 145 

plotting traverse by, 156 
Series method of measuring angles, 200 
Set-square : 

for drawing, 83 

large wooden, 21 
Setting out : 

brdge, 311 

building work, 309 

curves, 292 
Setting out parallel line : 

with chain, etc., 71 

with theodolite, 209 
Setting out railway, 264, 290, 304 
Setting out right angle : 

from a point to a chain line, 71, 208 

with chain, etc., 69 

with optical square, 18 
Setting up level, 237 
Setting up theodolite, 134, 136 
Sewer : 

locating route, 280 

longitudinal section of, 262 
Sextant, 114 
Shifting head, 136 
Shrinkage of plan, 327 
Signals : 

in levelling, 238 

in ranging lines, 27 

in triangulation, 197 
Simpson's rule, 322 
Sliding knot, 135 
Solid bodies, volumes of, 331 
Solid staff, 234 
Sopwith staff, 235 
Spikes, 15, 59 
Spirit level, 226 
Square : 

optical, 16 

prism, 20 

M'ooden, 21 



Square measure, 313 
Staff: 

cross, 15 

levelling, 233 

offset, 12 
Standard for testing chain, etc., 31 
Standards of theodolite, 125 
Stand for theodolite, 122 
Stations, survey, 59, 145, 181, 195, 202* 
Straight edge, 84 
Straight lines, ranging : 

by eye, 26 

by theodolite, 140 

past obstacles, 72, 209 
Surface of ground, methods of show- 
ing. 1 
Surveying : 

chain, 10 

compass, 101, 111 

difficulties, 9 

far side of river, 78 

field book, 49 

geometrical principles, 2 

plane, 1 

pond, wood, etc., 78 

traverse, 143 

triangulation, 186 
Survey line : 

chaining, 31 

examples of, 52 

fixing objects by offsets, 40, 43," 44 

fixing objects by ties, 40 

points to be fixed, 43 

running a, 48 
Survey lines, arrangement of, 55 112,. 

143 
Surveyor's compass, 106 
Surveyor's sextant, 114 
Survey party, 26 
Survey plan, example of, 97, 263 
Survey stations, 59, 145, 181, 195, 202- 



Tangent method of laying of 

on ground, 72 

on paper, 155 
Tape, linen and steel, 12 
Telescope of theodolite, 126 
Testing : 

band or chain, 31 

optical square, 19 

parallel ruler, 87 

set-square, 84 

sextant, 117 

straight edge, 84 
Theodolite : 

adjustment of, 348 

compass on, 109, 125 



INDEX 



36S 



Theodolite — continued. 
diaphragm, 128 
elements of, 120 
graduated circle, 124 
parallel plates, 123 
standards, 125 
support or stand, 122 
telescope, 126 
three screw levelling, 123 
trough compass, 125 
use of, 133 
vernier, 128 
vernier plate, 124 
vertical circle, 125 
Thread, stretched to give straight line, 

85 
Three point problem, 217 
Three screw levelling, 123, 231 
Ties, 40, 91 
Tinting, 99 

Trapezoidal rule for areas, 321 
Traverse polygon : 
area of, 326 
omitted side, 213 
two omitted measurements, 216 
Traverse survey : 

alteration of bearings for new axes, 

182 
checks on unclosed, 162 
closing error, 164, 184 
fixing stations, 145 
laying out lines, 144 
limits of error, 184 
methods of reading angles and 

bearings, 145 
plotting by angle and distance, 153 
plotting by co-ordinates, 166 
with compass, 112 
with theodolite, 143 
Traverse survey lines : 

connecting from one sheet to 

another, 183, 214 
correcting for errors in chainage, 

177 
plotting by co-ordinate methods, 

166 
plotting by separate angle methods, 

156 
plotting by whole circle bearing 
methods, 158 



Traverse survey lines — continued. 

plotting rectangle, 181 

weighting, 177 
Triangles : 

ill-conditioned, 55 

plotting main, 89, 202 
Triangulation : 

base line, 190 

broken base line, 192 

calculation of sides, 188, 354 

closing error in, 190 

enlarging base line, 193 

field work, 190 

marking stations, 197 

measuring angles, 198 

plotting stations, 202 

precautions in, 188 

purpose of, 186 

selecting stations, 195 
Trigonometric formulae, 353 
Trigonometric levelling, 220 
Tripod, 122, 231 
Trough compass, 125 
Troughton and Simms' level, 228, 23S 
True bearing, 105 



Vernier, 128, 130, 351 
Vernier p^te, 124 
Vertical circle, 125 
Volumes of solid bodies, 331 



W. 

Water level, 225 
Whole circle bearings : 

measuring, 148 

plotting traverse survey lines by, 
158 
Wood, surveying, 78 
Wooden measuring rods, 14 
Wooden pegs, 59 
Wooden set-square, 21, 42 
Wye level, 228, 233 
Wye level, adjustment of, 347 



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Bedell, F., and Pierce, C. A. Direct and Alternating Current Manual. 

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Begtrup, J. The Slide Valve 8vo, *2 3 

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Bender, C. E. Continuous Bridges. (Science Series No. 26.).. . . i6mo, o 50 

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Bennett, H. G. The Manufacture of Leather 8vo, *4 50 

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Bernthsen, A. A Text - book of Organic Chemistry. Trans, by G. 

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Bertin, L. E. Marine Boilers. Trans, by L. S. Robertson 8vo, 5 00 

Beveridge, J. Papermaker's Pocket Book nmo, *4 00 

Binnie, Sir A. Rainfall Reservoirs and Water Supply 3vo, *3 00 

Binns, C. F. Ceramic Technology 8vo, *5 00 

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Bligh, W. G. The Practical Design of Irrigation Works 8vo, *6 00 

Bloch, L. Science of Illumination. Trans, by W. C. Clinton 8\o, *2 50 

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fc?o, *7 50 

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Bodmer, G. R. Hydraulic Motors and Turbines i2mo, 5 00 

Boileau, J. T. Traverse Tables 8vo, 5 00 

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Bourgougnon, A. Physical Problems. (Science Series No. 113.) . i6mo, 050 
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Bow, R. H. A Treatise on Bracing 8vo, 1 50 

Bowie, A. J., Jr. A Practical Treatise on Hydraulic Mining 8vo, 5 00 

Bowker, W. R. Dynamo, Motor and Switchboard Circuits 8vo, *2 50 

Bowles, O. Tables of Common Rocks. (Science Series No. 125.). i6mo, 050 

Bowser, E. A. Elementary Treatise on Analytic Geometry nmo, 1 75 

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Design of Marine Engines and Auxiliaries (In Press.) 

Brainard, F. R. The Sextant. (Science Series No. 101.) i6mo, 

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Brew, W. Three-Phase Transmission 8vo, *2 00 

Briggs, R., and Wolff, A. R. Steam-Heating. (Science Series No. 

67.) i6mo, o 50 

Bright, C. The Life Story of Sir Charles Tilson Bright 8vo, *4 50 

Brislee, T. J. Introduction to the Study of Fuel. (Outlines of Indus- 
trial Chemistry. ) 8vo, *3 00 

Broadfoot, S. K. Motors, Secondary Batteries. (Installation Manuals 

Series.) nmo, *o 75 

Broughton, H. H. Electric Cranes and Hoists *g 00 

Brown, G. Healthy Foundations. (Science Series No. 80.) i6mo, o 50 



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Brown, Wm. N. The Art of Enamelling on Metal i2mo, *i oo 

• Handbook on Japanning and Enamelling i2mo, *i 50 

House Decorating and Painting nmo, *i 50 

History of Decorative Art i2mo, *i 25 

Brown, Wm. N. Dipping, Burnishing, Lacquering and Bronzing 

Brass Ware i2mo, *i 00 

Workshop Wrinkles 8vo, *i 00 

Browne, C. L. Fitting and Erecting of Engines 8vo, *i 50 

Browne, R. E. Water Meters. (Science Series No. 81.) i6mo, o 50 

Bruce, E. M. Pure Food Tests nmo, *i 25 

Bruhns, Dr. New Manual of Logarithms 8vo, cloth, 2 00 

half morocco, 2 50 
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Buel, R. H. Safety Valves. (Science Series No. 21.) i6mo, o 50 

Burns, D. Safety in Coal Mines nmo, *i 00 

Burstall, F. W. Energy Diagram for Gas. With Text 8vo, 1 50 

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Burton, F. G. Engineering Estimates and Cost Accounts i2mo, *i 50 

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Byers, H. G., and Knight, H. G. Notes on Qualitative Analysis .... 8vo, *i 50 

Cain, W. Brief Course in the Calculus i2mo, *i 75 

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Maximum Stresses. (Science Series No. 38.) i6mo, o 5c 

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Campin, F. The Construction of Iron Roofs 8vo, 2 00 

Carpenter, F. D. Geographical Surveying. (Science Series No. 37.).i6mo, 

Carpenter, R. C, and Diederichs, H. Internal Combustion Engines.. 8vo, *5 00 

Carter, E. T. Motive Power and Gearing for Electrical Machinery. 8vo, *5 00 

Carter, H. A. Ramie (Rhea), China Grass i2;no, *2 00 

Carter, H. R. Modern Flax, Hemp, and Jute Spinning 8vo, *3 00 

Cary, E. R. Solution of Railroad Problems with the Slide Rule i6mo, *i 00 

Cathcart, W. L. Machine Design. Part I. Fastenings 8vo, *3 00 

Cathcart, W. L., and Chaffee, J. I. Elements of Graphic Statics . . 8vo, *3 00 

Short Course in Graphics nmo, 1 50 

Caven, R. M., and Lander, G. D. Systematic Inorganic Ch.emisU7.12mo, *2 00 

Chalkley, A. P. Diesel Engines 8vo, *3 00 

Chambers' Mathematical Tables 8vo, 1 75 

Chambers, G. F. Astronomy i6mo, *i 50 

Charpentier, P. Timber 8vo, *6 00 



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How to Use Water Power nmo, *i 00 

Gyrostatic Balancing 8vo, *i 00 

Child, C. D. Electric Arc 8vo, *2 00 

Child, C. T. The How and Why of Electricity nmo, 1 00 

Christian, M. Disinfection and Disinfectants. Trans, by Chas. 

Salter i2mo, 200 

Christie, W. W. Boiler-waters, Scale, Corrosion, Foaming 8vo, *3 00 

Chimney Design and Theory 8vo, *3 00 

Furnace Draft. (Science Series No. 123.) i6mo, o 50 

Water: Its Purification and Use in the Industries 8vo, *2 00 

Church's Laboratory Guide. Rewritten by Edward Kinch 8vo, *2 50 

Clapperton, G. Practical Papermaking 8vo, 2 50 

Clark, A. G. Motor Car Engineering. 

Vol. I. Construction *3 00 

Vol. II. Design (In Press.) 

Clark, C. H. Marine Gas Engines i2mo, *i 50 

Clark, D. K. Fuel: Its Combustion and Economy i2mo, 1 50 

Clark, J. M. New System of Laying Out Railway Turnouts i2mo, 1 00 

Clarke, J. W., and Scott, W. Plumbing Practice. 

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Vol. II. Sanitary Plumbing and Fittings (/;/ Press.) 

Vol. III. Practical Lead Working on Roofs (In Press.) 

Clausen-Thue, W. ABC Telegraphic Code. Fourth Edition . . . i2mo, *5 00 

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The A 1 Telegraphic Code 8vo, *7 50 

Clerk, D., and Idell, F. E. Theory of the Gas Engine. (Science Series 

No. 62.) i6mo, o 50 

Clevenger, S. R. Treatise on the Method of Government Surveying. 

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Clouth, F. Rubber, Gutta-Percha, and Balata 8vo, *5 00 

Cochran, J. Concrete and Reinforced Concrete Specifications 8vo, *2 50 

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Coffin, J. H. C. Navigation and Nautical Astronomy nmo, *3 50 

Colburn, Z., and Thurston, R. H. Steam Boiler Explosions. (Science 

Series No. 2.) i6mo, 50 

Cole, R. S. Treatise on Photographic Optics i2mo, 1 50 

Coles-Finch, W. Water, Its Origin and Use 8vo, *5 00 

Collins, J. E. Useful Alloys and Memoranda for Goldsmiths, Jewelers. 

i6mo, o 50 

Collis, A. G. High and Low Tension Switch-Gear Design 8vo, *3 50 

Switchgear. (Installation Manuals Series. ) i2mo, *o 50 

Constantine, E. Marine Engineers, Their Qualifications and Duties. . 8vo, *2 00 

Coombs, H. A. Gear Teeth. (Science Series No. 120.) i6mo, o 50 

Cooper, W. R. Primary Batteries 8vo, *4 00 

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Part II *2 50 

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Corfield, W. H. Dwelling Houses. (Science Series No. 50.) .... i6mo, o 50 

Water and Water-Supply. (Science Series No. 17.) i6mo, o 50 

Cornwall, H. B. Manual of Blow-pipe Analysis 8vo, *2 50 

Courtney, C. F. Masonry Dams 8vo, 3 50 

Cowell, W. B. Pure Air, Ozone, and Water nmo, *2 00 

Craig, T. Motion of a Solid in a Fuel. (Science Series No. 49.) . i6mo, o 50 

Wave and Vortex Motion. (Science Series No. 43.) i6mo, o 50 

Cramp, W. Continuous Current Machine Design 8vo, *2 50 

Creedy, F. Single Phase Commutator Motors 8vo, *2 00 

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Vol. II. Distributing Systems and Lamps 

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Crocker, F. B., and Wheeler, S. S. The Management of Electrical Ma- 
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Cross, C. F., Bevan, E. J., and Sindall, R. W. Wood Pulp and Its Applica- 
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Crosskey, L. R. Elementary Perspective 8vo, 1 00 

Crosskey, L. R., and Thaw, J. Advanced Perspective 8vo, 

Culley, J. L. Theory of Arches. (Science Series No. 87.) i6mo, 

Dadourian, H. M. Analytical Mechanics nmo, 

Danby, A. Natural Rock Asphalts and Bitumens 8vo, 

Davenport, C. The Book. (Westminster Series.) 8vo, 

Davey, N. The Gas Turbine 8vo, 

Davies, D. C. Metalliferous Minerals and Mining 8vo, 

Earthy Minerals and Mining 8vo, 

Davies, E. H. Machinery for Metalliferous Mines 8vo, 

Davies, F. H. Electric Power and Traction 8vo, 

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Day, C. The Indicator and Its Diagrams i2mo, 

Deerr, N. Sugar and the Sugar Cane 8vo, 

Deite, C. Manual of Soapmaking. Trans, by S. T. King 4to, 

DelaCoux, H. The Industrial Uses of Water. Trans, by A. Morris. 8vo, 

Del Mar, W. A. Electric Power Conductors 8vo, 

Denny, G. A. Deep-level Mines of the Rand 4to, *io 00 

- — - Diamond Drilling for Gold *5 00 

De Roos, J. D. C. Linkages. (Science Series No. 47.) i6mo, o 50 

Derr, W. L. Block Signal Operation Oblong nmo, *i 50 

■ Maintenance-of-Way Engineering (In Preparation.) 

Desaint, A. Three Hundred Shades and How to Mix Them 8vo, 

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Devey, R. G. Mill and Factory Wiring. (Installation Manuals Series.) 

i2mo, 

Dibdin, W. J. Public Lighting by Gas and Electricity 8vo, 

Purification of Sewage and Water 8vo, 



I 


50 





50 


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lichmann, Carl. Basic Open-Hearth Steel Process i2mo, *3 50 

Bieterich, K. Analysis of Resins, Balsams, and Gum Resins, 8vo, *3 00 

Dinger, Lieut. H. C. Care and Operation of Naval Machinery . . . i2mo, *2 00 
Dixon, D. B. Machinist's and Steam Engineer's Practical Calculator. 

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Doble, W. A. Power Plant Construction on the Pacific Coast (In Press.) 

Dommett, W. E. Motor Car Mechanism nmo, *i 25 

Dorr, B. F. The Surveyor's Guide and Pocket Table-book. 

i6mo, morocco, 

Down, P. B. Handy Copper Wire Table i6mo, 

Draper, C. H. Elementary Text-book of Light, Heat and Sound . . i2mo, 

Heat and the Principles of Thermo-dynamics i2mo, 

Dubbel, H. High Power Gas Engines 8vo, 

Duckwall, E. W. Canning and Preserving of Food Products 8vo, 

Dumesny, P., and Noyer, J. Wood Products, Distillates, and Extracts. 

8vo, 
Duncan, W. G., and Penman, D. The Electrical Equipment of Collieries. 

8vo, 
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nmo, 
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Dwight, H. B. Transmission Line Formulas 8vo, 

Dyson, S. S. Practical Testing of Raw Materials 8vo, 

Dyson, S. S., and Clarkson, S. S. Chemical Works 8vo, 

Eccles, R. G., and Duckwall, E. W. Food Preservatives . . . 8vo, paper, o 50 
Eck, J. Light, Radiation and Illumination. Trans, by Paul Hogner, 

8vo, 

Eddy, H. T. Maximum Stresses under Concentrated Loads 8vo, 

Edelman, P. Inventions and Patents i2mo. (In Press.) 

Edgcumbe, K. Industrial Electrical Measuring Instruments 8vo, 

Edler, R. Switches and Switchgear. Trans, by Ph. Laubach. . .8vo, 

Eissler, M. The Metallurgy of Gold 8vo, 

The Hydrometallurgy of Copper 8vo, 

The Metallurgy of Silver 8vo, 

— — The Metallurgy of Argentiferous Lead 8vo, 

A Handbook on Modern Explosives 8vo, 

Ekin, T. C. Water Pipe and Sewage Discharge Diagrams folio, 

Eliot, C. W., and Storer, F. H. Compendious Manual of Qualitative 

Chemical Analysis nmo, 

Ellis, C. Hydrogenation of Oils 8vo, 

Ellis, G. Modern Technical Drawing 8vo, 

Ennis, Wm. D. Linseed Oil and Other Seed Oils 8vo, 

Applied Thermodynamics 8vo, 

Flying Machines To-day nmo, 

Vapors for Heat Engines , . nmo, 

Erfurt, J. Dyeing of Paper Pulp. Trans, by J. Hubner 8vo, 

Ermen, W. F. A. Materials Used in Sizing 8vo, 

Evans, C. A. Macadamized Roads (In Press.) 



2 


00 


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00 


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00 


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Ewing, A. J. Magnetic Induction in Iron 8vo, *4 ot> 

Fairie, J. Notes on Lead Ores nmo, *i oo 

Notes on Pottery Clays i2mo, *i 50 

Fairley, W., and Andre, Geo. J. Ventilation of Coal Mines. (Science 

Series No. 58.) i6mo, o 50 

Fair weather, W. C. Foreign and Colonial Patent Laws 8vo, *3 00 

Fanning, J. T. Hydraulic and Water-supply Engineering 8vo, *5 00 

Fauth, P. The Moon in Modern Astronomy. Trans, by J. McCabe. 

8vo, *2 00 

Fay, I. W. The Coal-tar Colors r 8vo, *4 00 

Fernbach, R. L. Glue and Gelatine 8vo, *3 00 

Chemical Aspects of Silk Manufacture. i2mo, *i 00 

Fischer, E. The Preparation of Organic Compounds. Tram, by R. V. 

Stanford 1 2mo, *i 25 

Fish, J. C. L. Lettering of Working Drawings Oblong 8vo, 1 00 

Fisher, H. K. C, and Darby, W. C. Submarine Cable Testing 8vo, *3 50 

Fleischmann, W. The Book of the Dairy. Trans, by C. M. Aikman. 

8vo, 4 00 
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Vol. II. The Utilization of Induced Currents *5 00 

Fleming, J. A. Propagation of Electric Currents 8vo, *3 00 

Centenary of the Electrical Current 8vo, *o 50 

Electric Lamps and Electric Lighting 8vo, *3 00 

Electrical Laboratory Notes and Forms 4to, *5 oo 

• A Handbook for the Electrical Laboratory and Testing Room. Two 

Volumes 8vo, each, *5 oo 

Fleury, P. Preparation and Uses of White Zinc Paints 8vo, *2 50 

Fleury, H. The Calculus Without Limits or Infinitesimals. Trans, by 

C. O. Mailloux (In Press.) 

Flynn, P. J. Flow of Water. (Science Series No. 84.) 121110, o 50 

Hydraulic Tables. (Science Series No. 66.) i6mo, o 50 

Foley, N. British and American Customary and Metric Measures . . folio, *3 oo 

Forgie, J. Shield Tunneling 8vo. (In Press. ) 

Foster, H. A. Electrical Engineers' Pocket-book. (Seventh Edition.) 

i2mo, leather, g 00 

Engineering Valuation of Publb Utilities and Factories 8vo, *3 00 

Handbook of Electrical Cost Data 8vo (In Press.) 

Foster, Gen. J. G. Submarine Blasting in Boston (Mass.) Harbor 4to, 3 53 

Fowle, F. F. Overhead Transmission Line Crossings i2mo, *i 50 

The Solution of Alternating Current Problems 8vo (In Press.) 

Fox, W. G. Transition Curves. (Science Series No. no.) i6mo, o 50 

Fox, W., and Thomas, C. W. Practical Course in Mechanical Draw- 
ing i2mo, 1 25 

Foye, J. C. Chemical Problems. (Science Series No. 69.) i6mo, o 50 

Handbook of Mineralogy. (Science Series No. 86.) i6mo, o 50 

Francis, J. B. Lowell Hydraulic Experiments 4to, 15 00 

Franzen, H. Exercises in Gas Analysis i2mo, *i 00 



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Freudemacher, P. W. Electrical Mining Installations. (Installation 

Manuals Series.) i2mo, *i oo 

Frith, J. Alternating Current Design 8vo, *2 oo 

Fritsch, J. Manufacture of Chemical Manures. Trans, by D. Grant. 

8vo, *4 oo 

Frye, A. I. Civil Engineers' Pocket-book i2mo, leather, *5 oo 

Fuller, G. W. Investigations into the Purification of the Ohio River. 

4to, *io oo 

Furnell, J. Paints, Colors, Oils, and Varnishes 8vo. *i oo 

Gairdner, J. W. I. Earthwork 8vo {In Press.) 

Gant, L. W. Elements of Electric Traction 8vo, *2 50 

Garcia, A. J. R. V. Spanish-English Railway Terms 8vo, *4 50 

Garforth, W. E. Rules for Recovering Coal Mines after Explosions and 

Fires i2mo, leather, 1 50 

Gaudard, J. Foundations. (Science Series No. 34.) i6mo, 050 

Gear, H. B., and Williams, P. F. Electric Central Station Distribution 

Systems 8vo, *3 00 

Geerligs, H. C. P. Cane Sugar and Its Manufacture 8vo, *5 00 

World's Cane Sugar Industry 8vo, *5 00 

Geikie, J. Structural and Field Geology 8vo, *4 00 

■ Mountains. Their Growth, Origin and Decay 8vo, *4 00 

The Antiquity of Man in Europe 8vo, *3 00 

Georgi, F., and Schubert, A. Sheet Metal Working. Trans, by C. 

Salter 8vo, 3 00' 

Gerber, N. Analysis of Milk, Condensed Milk, and Infants' Milk-Food. 8vo, 1 25 
Gerhard, W. P. Sanitation, Watersupply and Sewage Disposal of Country 

Houses i2mo, *2 oo 

Gas Lighting (Science Series No. in.) i6mo, o 50 

Household Wastes. (Science Series No. 97.) i6mo, o 50 

House Drainage. (Science Series No. 63.) i6mo, o 50 

Gerhard, W- P. Sanitary Drainage of Buildings. (Science Series No. 93.) 

i6mo, o 50 

Gerhardi, C. W. H. Electricity Meters 8vo, *4 00 

Geschwind, L. Manufacture of Alum and Sulphates. Trans, by C. 

Salter 8vo, *5 00 

Gibbs, W. E. Lighting by Acetylene i2mo, *i 50 

Physics of Solids and Fluids. (Carnegie Technical School's Text- 
books.) *i 50 

Gibson, A. H. Hydraulics and Its Application 8vo, *5 00 

Water Hammer in Hydraulic Pipe Lines i2mo, * 2 00 

Gilbreth, F. B. Motion Study nmo, *2 00 

Primer of Scientific Management i2mo, *i 00 

Gillmore, Gen. Q. A. Limes, Hydraulic Cements ard Mortars 8vo, 4 oo 

• Roads, Streets, and Pavements nmo, 2 00 

Golding, H. A. The Theta-Phi Diagram nmo, *i 25 

Goldschmidt, R. Alternating Current Commutator Motor 8vo, *3 00 

Goodchild, W. Precious Stones. (Westminster Series.) 8vo, *2 00 

Goodeve, T. M. Textbook on the Steam-engine nmo, 2 00 

Gore, G. Electrolytic Separation of Metals 8vo, *3 50 



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Gould, E. S. Arithmetic of the Steam-engine nmo, 

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High Masonry Dams. (Science Series No. 22.) i6mo, 

Practical Hydrostatics and Hydrostatic Formulas. (Science Series 

No. 117.) i6mo, 

Gratacap, L. P. A Popular Guide to Minerals 8vo, 

Gray, J. Electrical Influence Machines nmo, 

Marine Boiler Design nmo, 

Greenhill, G. Dynamics of Mechanical Flight 8vo, 

Greenwood, E. Classified Guide to Technical and Commercial Books. 8vo, 

Gregorius, R. Mineral Waxes. Trans, by C. Salter nmo, 

Griffiths, A. B. A Treatise on Manures nmo, 

Dental Metallurgy 8vo, 

Gross, E. Hops 8vo, 

Grossman, J. Ammonia and Its Compounds nmo, 

Groth, L. A. Welding and Cutting Metals by Gases or Electricity. 

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Gruner, A. Power-loom Weaving 8vo, 

Giildner, Hugo. Internal Combustion Engines. Trans, by H. Diederichs. 

4to, 

Gunther, C. 0. Integration nmo, 

Gurden, R. L. Traversa Tables folio, half morocco, 

Guy, A, E. Experiments on the Flexure of Beams 8vo, 

Haeder, H. Handbook on the Steam-engine. Trans, by H. H. P. 

Powles nmo, 

Hainbach, R. Pottery Decoration. Trans, by C. Salter nmo, 

Haenig, A. Emery and Emery Industry 8vo, 

Hale, W. J. Calculations of General Chemistry nmo, 

Hall, C. H. Chemistry of Paints and Paint Vehicles nmo, 

Hall, G. L. Elementary Theory of Alternate Current Working. .. .8vo, 

Hall, R. H. Governors and Governing Mechanism nmo, 

Hall, W. S. Elements of the Differential and Integral Calculus 8vo, 

Descriptive Geometry 8vo volume and a 4to atlas, 

Haller, G. F., and Cunningham, E. T. The Tesla Coil nmo, 

Halsey, F. A. Slide Valve Gears nmo, 

The Use of the Slide Rule. (Science Series No. 114.) i6mo, 

Worm and Spiral Gearing. (Science Series No. 116.) i6mo, 

Hamilton, W. G. Useful Information for Railway Men i6mo, 

Hammer, W. J. Radium and Other Radio-active Substances 8vo, 

Hancock, H. Textbook of Mechanics and Hydrostatics 8vo, 

Hancock, W. C. Refractory Materials. (Metallurgy Series.) (/;; Press.) 

Hardy, E. Elementary Principles of Graphic Statics nmo, *i 50 

Harris, S. M. Practical Topographical Surveying (In Press.) 

Harrison, W. B. The Mechanics' Tool-book nmo, 

Hart, J. W. External Plumbing Work 8vo, 

Hints to Plumbers on Joint Wiping 8vo, 

Principles of Hot Water Supply 8vo 

Sanitary Plumbing and Drainage 8vo, 



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D. VAN* NOSTRAND CO.'S SHORT TITLE CATALOG 



13 



Haskins, C. H. The Galvanometer and Its Uses i6mo, 1 50 

Hatt, J. A. H. The Colorist square nmo, *i 50 

Hausbrand, E. Drying by Means of Air and Steam. Trans, by A. C. 

Wright nmo, *2 00 

Evaporating, Condensing and Cooling Apparatus. Trans, by A. C. 

Wright 8vo, *5 00 

Hausner, A. Manufacture of Preserved Foods and Sweetmeats. Trans. 

by A. Morris and H. Robson 8vo, 

Hawke, W. H. Premier Cipher Telegraphic Code 4to, 

100,000 Words Supplement to the Premier Code 4to, 

Hawkesworth, J. Graphical Handbook for Reinforced Concrete Design. 

4to, 

Hay, A. Alternating Currents 8vo, 

Electrical Distributing Networks and Distributing Lines 8vo, 

Continuous Current Engineering 8vo, 

Hayes, H. V. Public Utilities, Their Cost New and Depreciation. . .8vo, 

Heap, Major D. P. Electrical Appliances 8vo, 

Heather, H. J. S. Electrical Engineering 8vo, 

Heaviside, O. Electromagnetic Theory. Vols. I and II ... . 8vo, each, 

Vol. Ill 8vo, 

Heck, R. C. H. The Steam Engine and Turbine 8vo, 

Steam-Engine and Other Steam Motors. Two Volumes. 

Vol. I. Thermodynamics and the Mechanics 8vo, 

Vol. II. Form, Construction, and Working 8vo, 

Notes on Elementary Kinematics 8vo, boards, 

Graphics of Machine Forces 8vo, boards, 

Hedges, K. Modern Lightning Conductors 8vo, 

Heermann, P. Dyers' Materials. Trans, by A. C. Wright nmo, 

Hellot, Macquer and D'Apligny. Art of Dyeing Wool, Silk and Cotton. 8vo, 

Henrici, 0. Skeleton Structures 8vo, 

Hering, D. W. Essentials of Physics for College Students 8vo, 

Hering-Shaw, A. Domestic Sanitation and Plumbing. Two Vols.. .8vo, 

Hering-Shaw, A. Elementary Science 8vo, 

Herrmann, G. The Graphical Statics of Mechanism. Trans, by A. P. 

Smith i2mo, 

Herzfeld, J. Testing of Yarns and Textile Fabrics 8vo, 

Hildebrandt, A. Airships, Past and Present 8vo, 

Hildenbrand, B. W. Cable-Making. (Science Series No. 32.). .. .i6mo, 

Hilditch, T. P. A Concise History of Chemistry nmo, 

Hill, J. W. The Purification of Public Water Supplies. New Edition. 

(In Press.) 

Interpretation of Water Analysis (In Press.) 

Hill, M. J. M. The Theory of Proportion 8vo, 

Hiroi, I. Plate Girder Construction. (Science Series No. g5.)...i6mo, 

Statically-Indeterminate Stresses nmo, 

Hirshfeld, C. F. Engineering Thermodynamics. (Science Series No. 45.) 

i6mo, 

Hobart, H. M. Heavy Electrical Engineering 8vo, 

Design of Static Transformers nmo, 

Electricity Svo, 

■ Electric Trains 8vo, 



-3 


00 


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00 


*5 


00 


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50 


*2 


50 


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5o 


*2 


5o 


*2 


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00 


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*2 


00 


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*I 


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*5 


00 


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00 


*3 


53 


*3 


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50 


*i 


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*2 


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*2 


00 


O 


50 


*4 50 


*2 


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*2 


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*2 


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i 4 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Hobart, H. M. Electric Propulsion of Ships 8vo, *2 oo 

Hobart, J. F. Hard Soldering, Soft Soldering and Brazing i2mo, *i oo 

Hobbs, W. R. P. The Arithmetic of Electrical Measurements iamo, o 50 

Hoff, J. N. Paint and Varnish Facts and Formulas i2mo, *i 50 

Hole, W. The Distribution of Gas 8vo, *7 50 

Holley, A. L. Railway Practice folio, 12 00 

Holmes, A. B. The Electric Light Popularly Explained. . . i2mo, paper, o 50 

Hopkins, N. M. Experimental Electrochemistry 8vo, *3 00 

Model Engines and Small Boats i2mo, 125 

Hopkinson, J., Shoolbred, J. N., and Day, R. E. Dynamic Electricity. 

(Science Series No. 71.) i6mo, o 50 

Horner, J. Metal Turning i2mo, 1 50 

Practical Ironf ounding 8vo, *2 00 

Plating and Boiler Making 8vo, 3 00 

Gear Cutting, in Theory and Practice 8vo, *3 cd 

Houghton, C. E. The Elements of Mechanics of Materials izmo, *2 oo 

Houllevigue, L. The Evolution of the Sciences 8vo, *2 00 

Houstoun, R. A. Studies in Light Production i2mo, 2 00 

Hovenden, F. Practical Mathematics for Young Engineers nmo, *i 00 

Howe, G. Mathematics for the Practical Man i2mo, *i 25 

Howorth, J. Repairing and Riveting Glass, China and Earthenware. 

8vo, paper, *o 50 

Hubbard, E. The Utilization of Wood-waste 8vo, *2 50 

Hiibner, J. Bleaching and Dyeing of Vegetable and Fibrous Materials. 

(Outlines of Industrial Chemistry.) 8vo, *5 00 

Hudson, 0. F. Iron and Steel. (Outlines of Industrial Chemistry. ).8vo, *2 00 

Humper, W. Calculation of Strains in Girdeis 12H10, 2 50 

Humphrey, J. C. W. Metallography of Strain. (Metallurgy Series.) 

( In Press.) 

Humphreys, A. C. The Business Features of Engineering Practice.. 8vo, *i 25 

Hunter, A. Bridge Work 8vo. ( In Press.) 

Hurst, G. H. Handbook of the Theory of Color 8vo, *2 50 

Dictionary of Chemicals and Raw Products 8vo, *3 00 

Lubricating Oils, Fats and Greases 8vo, *4 00 

Soaps 8vo, *5 00 

Hurst, G. H., and Simmons, W. H. Textile Soaps and Oils 8vo, *2 50 

Hurst, H. E., and Lattey, R. T. Text-book of Physics 8vo, *3 00 

Also published in three parts. 

Part I. Dynamics and Heat *i 25 

Part II. Sound and Light *i 25 

Part III. Magnetism and Electricity *i 50 

Hutchinson, R. W., Jr. Long Distance Electric Power Transmission. 

nmo, *3 00 
Hutchinson, R. W., Jr., and Thomas, W. A. Electricity in Mining. i2mo, 

{In Press.) 
Hutchinson, W. B. Patents and How to Make Money Out of Them. 

i2mo, 1 25 

Hutton, W. S. Steam-boiler Construction , 8vo, 6 00 

Practical Engineer's Handbook 8vo, 7 00 

The Works' Manager's Handbook 8vo, 6 00 



D VAN NOSTRAND CO.'S SHORT TITLE CATALOG 



15 






5^> 


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50 


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03 


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50 



Hyde, E. W. Skew Arches. (Science Series No. 15.) i6mo, o 50 

Hyde, F. S. Solvents, Oils, Gums, Waxes. 8vo, *2 00 

Induction Coils. (Science S3rios No. 53.) i6mo, 

Ingham, A. E. Gearing. A practical treatise 8vo, 

Ingle, H. Manual of Agricultural Chemistry 8vo, 

Inness, C. H. Problems in Machine Design i2mo, 

Air Compressors and Blowing Engines nmo, 

Centrifugal Pumps 121110, 

— — The Fan nmo, 

Isherwood, B. F. Engineering Precedents for Steam Machinery . . . 8vo, 
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Jacob, A., and Gould, E. S. On the Designing and Construction of 

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Jannettaz, E. Guide to the Determination of Rocks. Trans, by G. W. 

Plympton i2mo, 1 50 

Jehl, F. Manufacture of Carbons 8vo, *4 00 

Jennings, A. S. Commercial Paints and Painting. (Westminster Series.) 

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Jennison, F. H. The Manufacture of Lake Pigments 8vo, *3 00 

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Johnston, J. F. W., and Cameron, C. Elements of Agricultural Chemistry 

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Jones, H. C. Electrical Nature of Matter and Radioactivity nmo, 

New Era in Chemistry i2mo, 

Jones, M. W. Testing Raw Materials Used in Paint nmo, 

Jones, L., and Scard, F. I. Manufacture of Cane Sugar 8vo, 

Jordan, L. C. Practical Railway Spiral nmo, leather, 

Joynson, F. H. Designing and Construction of Machine Gearing . .8vo, 
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Kansas City Bridge 4to, 

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Keim, A. W. Prevention of Dampness in Buildings 8vo, 

Keller, S. S. Mathematics for Engineering Students. 1 21110, half leather. 

Algebra and Trigonometry, with a Chapter on Vectors *i 75 

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Plane and Solid Geometry *i . 25 

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Kemble, W. T., and Underhill, C. R. The Periodic Law and the Hydrogen 

Spectrum 8vo, paper, *o 50 



*0 


75 


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00 


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00 


*2 


00 


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50 


2 


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*5 


00 


6 


00 





50 


*2 


00 



j6 d. VAN NOSTRAND CO.'S short title catalog 

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Kennedy, A. B. W., Unwin, W. C, and Idell, F. E. Compressed Air. 

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Electrometallurgy. (Westminster Series.) 8vo, *2 00 

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Electro-Thermal Methods of Iron and Steel Production. .. .8vo, *3 00 

Kinzbrunner, C. Alternate Current Windings 8vo, *i 50 

Continuous Current Armatures 8vo, *i 50 

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Kirschke, A. Gas and Oil Engines i2tno, *i 25 

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Lassar-Cohn. Dr. Modern Scientific Chemistry. Trans, by M. M. 

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Lawson, W. R. British Railways. A Financial and Commercial 

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Lecky, S. T. S. " Wrinkles " in Practical Navigation 8vo, *8 00 

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Lemstrom, S. Electricity in Agriculture and Horticulture 8vo, *i 50 

Letts, E. A. Fundamental Problems in Chemistry 8vo, *2 00 

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Carbonization of Coal 8vo, *3 00 

Lewis, L. P. Railway Signal Engineering 8vo, *3 50 

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Bankers and Stockbrokers' Code and Merchants and Shippers' 

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100,000,000 Combination Code 8vo, *io 00 

Engineering Code 8vo, *i2 50 

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Livingstone, R. Design and Construction of Commutators 8vo, *2 25 

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Lobben, P. Machinists' and Draftsmen's Handbook 8vo, 2 50 

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Loring, A. E. A Handbook of the Electromagnetic Telegraph .... i6mo o 50 

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Marshall, W. J., and Sankey, H. R. Gas Engines. (Westminster Series.) 

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20 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

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Morgan, A. P. Wireless Telegraph Apparatus for Amateurs nmo, *i 50 

Moses, A. J. The Characters of Crystals 8vo, *2 00 

and Parsons, C. L. Elements of Mineralogy 8vo, *2 50 

Moss, S.A. Elementsof Gas Engine Design. (Science Series No. 121.) 1 6mo, 50 

The Lay-out of Corliss Valve Gears. (Science Series No. 1 19.) i6mo, o 50 

Mulford, A. C. Boundaries and Landmarks nmo, *i 00 

Mullin, J. P. Modern Moulding and Pattern-making nmo, 2 50 

Munby, A. E. Chemistry and Physics of Building Materials. (West- 
minster Series.) 8vo, *2 00 

Murphy, J. G. Practical Mining i6mo, 1 00 

Murphy, W. S. Textile Industries. Eight Volumes *2o 00 

Sold separately, each, :: 3 00 

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Naquet, A. Legal Chemistry 12010, 2 00 

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Recent Cotton Mill Construction nmo, 2 00 

Neave, G. B., and Heilbron, I. M. Identification of Organic Compounds. 

nmo, *i 25 

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Nerz, F. Searchlights. Trans, by C. Rodgers 8vo, *3 00 

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Newall, J. W. Drawing, Sizing and Cutting Bevel-gears 8vo, 1 50 

Nicol, G. Ship Construction and Calculations 8vo, *4 50 

Nipher, F. E. Theory of Magnetic Measurements nmo, 1 00 

Nisbet, H. Grammar of Textile Design 8vo, "3 00 

Nolan, H. The Telescope. (Science Series No. 51.) i6mo, o 50 

Noll, A. How to Wire Buildings nmo, 1 50 

North, H. B. Laboratory Experiments in General Chemistry nmo, *i 00 

Nugent, E. Treatise on Optics nmo, 1 50 

O'Connor, H. The Gas Engineer's Pocketbook nmo, leather, 3 50 

Petrol Air Gas nmo, *o 75 



D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 21 

Ohm, G. S., and Lockwood, T. D. Galvanic Circuit. Translated by 

William Francis. (Science Series No. 102.) i6mo, o 50 

Olsen, J. C. Text-book of Quantitative Chemical Analysis 8vo, *4 00 

Olsson, A. Motor Control, in Turret Turning and Gun Elevating. (U. S. 

Navy Electrical Series, No. 1.) nmo, paper, *o 50 

Ormsby, M. T. M. Surveying i2mo, 1 50 

Oudin, M. A. Standard Polyphase Apparatus and Systems 8vo, *3 00 

Cwen, D. Recent Physical Research 8vo, *i 50 

Pakes, W. C.C., and Nankivell, A. T. The Science of Hygiene . .8vo, *i 75 

Palaz, A. Industrial Photometry. Trans, by G. W. Patterson, Jr . . 8vo, *4 00 

Pamely, C. Colliery Manager's Handbook 8vo, *io 00 

Parker, P. A. M. The Control of Water 8vo, *5 00 

Parr, G. D. A. Electrical Engineering Measuring Instruments. .. .8vo, *3 50 

Parry, E. J. Chemistry of Essential Oils and Artificial Perfumes. . .8vo, *5 00 

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and Parry, E. Electrical Equipment of Tramways. . (In Press.) 

Parsons, S. J. Malleable Cast Iron 8vo, *2 50 

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Paterson, G. W. L. Wiring Calculations i2mo, *2 00 

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The Science of Color Mixing 8vo, *3 00 

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Peddie, R. A. Engineering and Metallurgical Books nmo, *i 50 

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Perry, J. Applied Mechanics 8vo, *2 50 

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22 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

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The Slide Rule nmo, 1 00 

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by H. B. Cornwall 8vo, *4 00 

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How to become an Engineer. (Science Series No. 100.) i6tno, 

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leather, 

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Potts, H. E. Chemistry of the Rubber Industry. (Outlines of Indus- 
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nmo, 

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Structures i2mo, *2 50 

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Pulsifer, W. H. Notes for a History of Lead 3vo, 4 00 

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Putsch, A. Gas and Coal-dust Firing 8vo, *3 00 

Pynchon, T. R. Introduction to Chemical Physics 8vo, 3 oa 






50 





5o 





50 





5^ 





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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 23 

Rafter G. W Mechanics of Ventilation. (Science Series No. 33.) . 16 mo, 

Potable Water. (Science Series No. 103.) i6rno, 

Treatment of Septic Sewage. (Science Series No. 118.) . . . i6mo, 

Rafter, G. W., and Baker, M. N. Sewage Disposal in the United States. 

4 to, 

Raikes, H. P. Sewage Disposal Works 8vo, 

Randall, P. M. Quartz Operator's Handbook nmo, 

Randau, P. Enamels and Enamelling 8vo, 

Rankine, W. J. M. Applied Mechanics 8vo, 

Civil Engineering , 8vo, 

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— — The Steam-engine and Other Prime Movers 8vo, 

Useful Rules and Tables 8vo, 

Rankine, W. J. M., and Bamber, E. F. A Mechanical Text-book.. . .8vo, 
Raphael, F. C. Localization of Faults in Electric Light and Power Mains. 

8vo, *3 00 

Rasch, E. Electric Arc Phenomena. Trans, by K. Tornberg 8vo, *i 00 

Rathbone, R. L. B. Simple Jewellery 8vo, *2 00 

Rateau, A. Flow of Steam through Nozzles and Orifices. Trans, by H. 

B. Brydon 8vo *i 50 

Rausenberger, F. The Theory of the Recoil of Guns 8vo, *4 50 

Rautenstrauch, W. Notes on the Elements of Machine Design. 8vo, boards, *i 50 
Rautenstrauch, W., and Williams, J. T. Machine Drafting and Empirical 

Design. 

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Part II. Empirical Design (In Preparation.) 

Raymond, E. B. Alternating Current Engineering i2mo, 

Rayner, H. Silk Throwing and Waste Silk Spinning 8vo, 

Recipes for the Color, Paint, Varnish, Oil, Soap and Drysaltery Trades . 8vo, 

Recipes for Flint Glass Making nmo, 

Redfern, J..B., and Savin, J. Bells, Telephones (Installation Manuals 

Series.) i6mo, 

Redgrove, H. S. Experimental Mensuration i2mo, 

Redwood, B. Petroleum. (Science Series No. 92.) . , i6mo, 

Reed, S. Turbines Applied to Marine Propulsion *5 00 

Reed's Engineers' Handbook 8vo, 

Key to the Nineteenth Edition of Reed's Engineers' Handbook . . 8vo, 

Useful Hints to Sea-going Engineers nmo, 

■ Marine Boilers nmo, 

Guide to the Use of the Slide Valve nmo, 

Reinhardt, C. W. Lettering for Draftsmen, Engineers, and Students. 

oblong 4to, boards, 1 00 

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Reiser, F. Hardening and Tempering of Steel. Trans, by A. Morris and 

H. Robson nmo, *2 50 

Reiser, N. Faults in the Manufacture of Woolen Goods. Trans, by A. 

Morris and H. Robson 8vo, *2 50 

Spinning and Weaving Calculations 8vo, *5 00 

Renwick, W. G. Marble and Marble Working 8vo, 5 00 



*2 


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60 



24 



D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 



Reynolds, 0., and Idell, F. E. Triple Expansion Engines. (Science 

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Rhead, G. F. Simple Structural Woodwork nmo, *i 00 

Rhodes, H. J. Art of Lithography 8vo, 3 50 

Rice, J. M., and Johnson, W. W. A New Method of Obtaining the Differ- 
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Richards, W. A. Forging of Iron and Steel (In Press.) 

Richards, W. A., and North, H. B. Manual of Cement Testing. . . . nmo, *i 50 

Richardson, J. The Modern Steam Engine 8vo, *3 50 

Richardson, S. S. Magnetism and Electricity nmo, *2 00 

Rideal, S. Glue and Glue Testing 8vo, *4 00 

Rimmer, E. J. Boiler Explosions, Collapses and Mishaps 8vo, *i 75 

Rings, F. Concrete in Theory and Practice nmo, *2 50 

Reinforced Concrete Bridges 4to, *5 00 

Ripper, W. Course of Instruction in Machine Drawing folio, *6 00 

Roberts, F. C. Figure of the Earth. (Science Series No. 79.) i6mo, o 50 

Roberts, J., Jr. Laboratory Work in Electrical Engineering 8vo, *2 00 

Robertson, L. S. Water-tube Boilers 8vo, 2 00 

Robinson, J. B. Architectural Composition 8vo, *2 50 

Robinson, S. W. Practical Treatise on the Teeth of Wheels. (Science 

Series No. 24.) i6mo, o 5c 

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Wrought Iron Bridge Members. (Science Series No. 60.) i6mo, o 50 

Robson, J. H. Machine Drawing and Sketching 8vo, T i 50 

Roebling, J. A. Long and Short Span Railway Bridges folio, 25 00 

Rogers, A. A Laboratory Guide of Industrial Chemistry nmo, *i 50 

Rogers, A., and Aubert, A. B. Industrial Chemistry 8vo, *5 00 

Rogers, F. Magnetism of Iron Vessels. (Science Series No. 30.). i6mo, o 5o 
Rohland, P. Colloidal and Crystalloidal State of Matter. Trans, by 

W. J. Britland and H. E. Potts nmo, *i 25 

Rollins, W. Notes on X-Light 8vo, *5 03 

Rollinson, C. Alphabets Cb!ong, nmo, *i 00 

Rose, J. The Pattern-makers' Assistant 8vo, 2 50 

— — Key to Engines and Engine-running nmo, 2 50 

Rose, T. K. The Precious Metals. (Westminster Series.) 8vo, *2 00 

Rosenhain, W. Glass Manufacture. (Westminster Series.) 8vo, *2 00 

Physical Metallurgy, An Introduction to. (Metallurgy Series.) 

8vo, (In Press.) 

Ross, W. A. Blowpipe in Chemistry and Metallurgy nmo, *2 00 

Roth. Physical Chemistry 8vo, *2 00 

Rouillion, L. The Economics of Manual Training 8vo, 2 00 

Rowan, F. J. Practical Physics of the Modern Steam-boiler 8vo, *3 00 

and Idell, F. E. Boiler Incrustation and Corrosion. (Science 

Series No. 27.) i6mo, 050 

Roxburgh, W. General Foundry Practice. (Westminster Series.) .8vo, *2 0: 

Ruhmer, E. Wireless Telephony. Trans, by J. Erskine-Murray. .8vo, *3 50 

Russell, A. Theory of Electric Cables and Networks 8vo, *3 00 

Sabine, R. History and Progress of the Electric Telegraph nmo, 1 25 

Saeltzer, A. Treatise on Acoustics nmo, 1 00 



D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 25 

Sanf ord, P. G. Nitro-explosives 8vo, 

Saunders, C. H. Handbook of Practical Mechanics i6mo, 

leather, 

Saunnier, C. Watchmaker's Handbook nmo, 

Sayers, H. M. Brakes for Tram Cars 8vo, 

Scheele, C. W. Chemical Essays 8vo, 

Scheithauer, W. Shale Oils and Tars 8vo, 

Schellen, H. Magneto-electric and Dynamo-electric Machines .... 8vo, 

Scherer, R. Casein. Trans, by C. Salter 8vo, 

Schidrowitz, P. Rubber, Its Production and Industrial Uses 8vo, 

Schindler, K. Iron and Steel Construction "Works nmo, 

Schmall, C. N. First Course in Analytic Geometry, Plane and Solid. 

i2mo, half leather, 
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Schmeer, L. Flow of Water 8vo, 

Schumann, F. A Manual of Heating and Ventilation. . . .izmo, leather, 

Schwarz, E. H. L. Causal Geology 8vo, 

Scirweizer, V. Distillation of Resins 8vo, 

Scott, W. W. Qualitative Analysis. A Laboratory Manual 8vo, 

Scribner, J. M. Engineers' and Mechanics' Companion. .i6mo, leather, 
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Organic Compounds 8vo, 

Searle, A. B. Modern Brickraaking 8vo, 

1 Cement, Concrete and Bricks 8vo, 

Searle, G. M. "Sumners' Method." Condensed and Improved. 

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Seaton, A. E. Manual of Marine Engineering 8vo 8 00 

Seaton, A. E., and Rounthwaite, H. M. Pocket-book of Marine Engi- 
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Seligman, R. Aluminum. (Metallurgy Series.) (In Press.) 

Sellew, W. H. Steel Rails 4 to, 

Senter, G. Outlines of Physical Chemistry i2mo, 

Text-book of Inorganic Chemistry i2mo, 

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Sever, G. F., and Townsend, F. Laboratory and Factory Tests in Elec- 
trical Engineering 8vo, 

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Lessons in Telegraphy i2mo, 

Sewell, T. Elements of Electrical Engineering 8vo, 

The Construction of Dynamos 8vo, 

Sexton, A. H. Fuel and Refractory Materials i2mo, 

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Alloys (Non-Ferrous) 8vo, 

— — The Metallurgy of Iron and Steel 8vo, 

Seymour, A. Practical Lithography 8vo, 

Modern Printing Inks 8vo, 



*4 


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*2 


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26 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Shaw, Henry S. H. Mechanical Integrators. (Science Series No. 83.) 

i6mo, 

Shaw, S. History of the Staffordshire Potteries 8vo, 

Chemistry of Compounds Used in Porcelain Manufacture. .. .8vo, 

Shaw, W. N. Forecasting Weather 8vo, 

Sheldon, S., and Hausmann, E. Direct Current Machines i2mo, 

Alternating Current Machines i2mo, 

Sheldon, S., and Hausmann, E. Electric Traction and Transmission 

Engineering 1 2mo, 

Sheriff, F. F. Oil Merchants' Manual i2mo, 

Shields, J. E. Notes on Engineering Construction i2mo, 

Shreve, S. H. Strength of Bridges and Roofs 8vo, 

Shunk, W. F. The Field Engineer i2mo, morocco, 

Simmons, W. H., and Appleton, H. A. Handbook of Soap Manufacture, 

8vo, 

Simmons, W. H., and Mitchell, C. A. Edible Fats and Oils 8vo, 

Simms, F. W. The Principles and Practice of Levelling 8vo, 

Practical Tunneling 8vo, 

Simpson, G. The Naval Constructor i2mo, morocco, 

Simpson, W. Foundations 8vo. (In Press.) 

Sinclair, A. Development of the Locomotive Engine. . . 8vo, half leather, 
Twentieth Century Locomotive 8vo, half leather, 

Sindall, R. W., and Bacon, W. N. The Testing of Wood Pulp 8vo, 

Sindall, R. W. Manufacture of Paper. (Westminster Series.). .. .8vo, 

Sloane, T. O'C. Elementary Electrical Calculations nmo, 

Smallwood, J. C. Mechanical Laboratory Methods. .. .i2mo, leather, 

Smith, C. A. M. Handbook of Testing, MATERIALS 8vo, 

Smith, C. A. M., and Warren, A. G. New Steam Tables 8vo, 

Smith, C. F. Practical Alternating Currents and Testing 8vo, 

Practical Testing of Dynamos and Motors 8vo, 

Smith, F. E. Handbook of General Instruction for Mechanics . . . nmo, 
Smith, H. G. Minerals and the Microscope 

Smith, J. C. Manufacture of Paint 8vo, *3 oc 

Paint and Painting Defects 

Smith, R. H. Principles of Machine Work nmo, 

— — Elements of Machine Work nmo, 

Smith, W. Chemistry of Hat Manufacturing nmo, 

Snell, A. T. Electric Motive Power 8vo, 

Snow, W. G. Pocketbook of Steam Heating and Ventilation. (In Press.) 
Snow, W. G., and Nolan, T. Ventilation of Buildings. Science Series 

No. 5.) i6mo, 

Soddy, F. Radioactivity 8vo, 

Solomon, M. Electric Lamps. (Westminster Series.) 8vo, 

Somerscales, A. N. Mechanics for Marine Engineers nmo, 

Mechanical and Marine Engineering Science 8vo, 

Sothern, J. W. The Marine Steam Turbine 8vo, 

Verbal Notes and Sketches for Marine Engineers 8vo, 

Sothern, J. W., and Sothern, R. M. Elementary Mathematics for 

Marine Engineers i2mo, 

Simple Problems in Marine Engineering Design nmo, 






50 


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53 


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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 



>-7 



Southcombe, J. E. Chemistry of the Oil Industries. (Outlines of In- 
dustrial Chemistry.) 8vo, *3 oo 

Soxhlet, D. H. Dyeing and Staining Marble. Trans, by A. Morris and 

H. Robson 8vo, *2 50 

Spang, H. W. A Practical Treatise on Lightning Protection nmo, 1 oo 

Spangenburg, L. Fatigue of Metals. Translated by S. H. Shreve. 

(Science Series No. 23.) i6mo, o 50 

Specht, G. J., Hardy, A. S., McMaster, J. B., and Walling. Topographical 

Surveying. (Science Series No. 72.) i6mo, o 50 

Speyers, C. L. Text-book of Physical Chemistry 8vo, *2 25 

Sprague, E. H. Hydraulics i2mo, 1 25 

Stahl, A. W. Transmission of Power. (Science Series No. 28.) . i6mo, 

Stahl, A. W., and Woods, A. T. Elementary Mechanism nmo, *2 00 

Staley, C, and Pierson, G. S. The Separate System of Sewerage. . .8vo, *3 00 

Standage, H. C. Leatherworkers' Manual 8vo, *3 50 

Sealing Waxes, Wafers, and Other Adhesives 8vo, *2 00 

Agglutinants of all Kinds for all Purposes nmo, *3 50 

Stanley, H. Practical Applied Physics (In Press.) 

Stansbie, J. H. Iron and Steel. (Westminster Series.) 8vo, *2 00 

Steadman, F. M. Unit Photography and Actinometry (In Press.) 

Steelier, G. E. Cork. Its Origin and Industrial Uses i2mo, 1 00 

Steinman, D. B. Suspension Bridges and Cantilevers. (Science Series 

No. 127.) o 50 

Stevens, H. P. Paper Mill Chemist . . . r i6mo, *2 50 

Stevens, J. S. Precision of Measurements (In Press.) 

Stevenson, J. L. Blast-Furnace Calculations i2mo, leather, *2 00 

Stewart, A. Modern Polyphase Machinery nmo, *2 00 

Stewart, G. Modern Steam Traps i2mo, *i 25 

Stiles, A. Tables for Field Engineers nmo, 1 00 

Stillman, P. Steam-engine Indicator i2mo, 1 00 

Stodola, A. Steam Turbines. Trans, by L. C. Loewenstein 8vo, *5 00 

Stone, H. The Timbers of Commerce 8vo, 3 50 

Stone, Gen. R. New Roads and Road Laws i2mo, 1 00 

Stopes, M. Ancient Plants 8vo, *2 00 

The Study of Plant Life 8vo, *2 00 

Stumpf, Prof. Una-Flow of Steam Engine 4to, *3 50 

Sudborough, J. J., and James, T. C. Practical Organic Chemistry.. i2mo, *2 00 

Suffling, E. R. Treatise on the Art of Glass Painting 8vo, *3 50 

Swan, K. Patents, Designs and Trade Marks. (Westminster Series.). 

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